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Punjab State Board PSEB 12th Class Chemistry Book Solutions Chapter 15 Polymers Textbook Exercise Questions and Answers.

PSEB Solutions for Class 12 Chemistry Chapter 15 Polymers

PSEB 12th Class Chemistry Guide Polymers InText Questions and Answers

Question 1.
Explain the terms polymer and monomer.
Answer:

• Polymer is a high molecular mass macromolecule consisting of repeating structural units derived from monomers.
• A monomer is a simple molecule capable of undergoing polymerisation and leading to the formation of the corresponding polymer.

Question 2.
What are natural and synthetic polymers? Give two examples of each type.
Answer:

1. Natural polymers are high molecular mass macromolecules and are found in plants and animals. For example, proteins and nucleic acids,
2. Synthetic polymers are man-made high molecular mass macromolecules. These include synthetic plastics, fibres and rubbers. For example, polythene and dacron.

Question 3.
Distinguish between the terms homopolymer and copolymer and give an example of each.
Answer:
Homopolymer: Polymers whose repeating structural units are derived from only one type of monomer units are called homopolymers. For example, polythene, PAN, Teflon, nylon-6, etc.

Copolymer: Polymers whose repeating structural units are derived from two or more types of monomer molecules are called copolymers. For example, Buna-S, Buna-N, nylon-6,6 polyester, bakelite, etc.

Question 4.
How do you explain the functionality of a monomer?
Answer:
The functionality of a monomer is the number of binding sites in a molecule. For example, the functionality of ethene, propene, styrene, acrylonitrile is one and that of 1, 3-butadiene, adipic acid, terephthalic acid, hexamethylenediamine is two.

Question 5.
Define the term polymerisation.
Answer:
The polymerisation is a process of formation of a high molecular mass polymer from one or more monomers by linking together a large number of repeating structural units through covalent bonds.

Question 6.

Answer:
is a homopolymer because it is obtained from a single monomer unit, NH₂-CHR-COOH.

Question 7.
In which classes, the polymers are classified on the basis of molecular forces?
Answer:
On the basis of molecular forces of attraction polymers are classified into the following classes :

• Elastomers
• Fibres
• Thermoplastic polymers and
• Thermosetting polymers.

Question 8.
How can you differentiate between addition and condensation polymerisation?
Answer:
(i) Addition polymerisation: In this, polymers are formed by the repeated addition of monomers molecules possessing double or triple bonds.
For example :

(ii) Condensation polymerisation: It is a process in which two or more bi-functional molecules undergo a series of condensation reactions with the elimination of some simple molecules and leading to the formation of polymers.

Question 9.
Explain the term copolymerisation and give two examples.
Answer:
When a mixture containing more than one monomeric species is allowed to polymerise, the product obtained is called a copolymer and the process is called copolymerisation. For example, Buna-S, a copolymer of 1,3-butadiene and styrene and Buna-N, a copolymer of 1,3-butadiene and acrylonitrile.

Question 10.
Write the free radical mechanism for the polymerisation of ethene.
Answer:
The mechanism of chain growth polymerisation of ethene of free radical mechanism is given below :
Step I. Chain initiation step :

Step II. Chain propagation step :

Step III. Chain terminating step :

Question 11.
Define thermoplastics and thermosetting polymers with two examples Of each.
Answer:
Thermoplastic polymers: These polymers are the linear or slightly. branched long-chain molecules capable of repeatedly softening on heating and hardening on cooling. These polymers possess intermolecular forces of attraction intermediate between elastomers and fibres. Some common examples are polythene, polystyrene, polyvinyl, etc.

Thermosetting polymers: These polymers are cross-linked or heavily branched molecules, which on heating undergo extensive cross-linking in moulds and again become infusible. These polymers cannot be reused. Some common examples are bakelite, urea-formaldehyde resins, etc.

Question 12.
Write the monomers used for getting the following polymers :
(i) Polyvinyl chloride
(ii) Teflon
(iii) Bakelite
Answer:

Question 13.
Write the name and structure of one of the common initiators used in free radical addition polymerisation.
Answer:
One common initiator used in free radical addition polymerisation is benzoyl peroxide. Its structure is given below:

Question 14.
How does the presence of double bonds in rubber molecules influence their structure and reactivity?
Answer:

• In natural rubber (cis- 1,4-polyisoprene), double bonds are located between C₂ and C₃ of isoprene units.
• The double bonds are reactive sites and helps in vulcanisation of natural rubber forming S-S linkages between chains.
• The cis-configuration about double bonds do not allow the chains to come closer and so there are weak intermolecular attractions between the chains. This leads to coiled structure and elasticity for natural rubber.

Question 15.
Discuss the main purpose of vulcanisation of rubber.
Answer:
Natural rubber becomes soft at high temperatures (> 335K) and brittle at low temperatures (< 283 K) and shows high water absorption capacity. It is soluble in non-polar solvents and is non-resistant to attack by oxidising agents. To improve upon these physical properties, a process of vulcanisation is carried out. This process consists of heating a mixture of raw rubber with sulphur and an appropriate additive at a temperature range between 373 K and 415 K. On vulcanisation, sulphur forms cross-links at the reactive sites of double bonds and thus the rubber gets stiffened.

Question 16.
What are the monomeric repeating units of Nylon-6 and Nylon-6,6?
Answer:

(ii) Nylon-6,6: 1,6-Hexamethylenediamine and adipic acid.

Question 17.
Write the names and structures of the monomers of the following polymers:
(i) Buna-S,
(ii) Buna-N,
(iii) Dacron,
(iv) Neoprene
Answer:
The names and structures of monomers are :

Question 18.
Identify the monomer in the following polymeric structures :

Answer:
The monomers forming the polymer are :
(i) Decanoic acid HOOC-(CH₂)₈-COOH and Hexamethylenediamine H₂N-(CH₂)₆-NH₂.
(ii)

Question 19.
How is dacron obtained from ethylene glycol and terephthalic acid?
Answer:
Dacron is a synthetic condensation polymer which has ester group in the polymer chain. Terylene was also known by the name.

Question 20.
What is a biodegradable polymer? Give an example of a biodegradable aliphatic polyester.
Answer:
A polymer that can be decomposed by bacteria is called a biodegradable polymer.
Poly-β-hydroxybutyrate-CO-β-hydroxy valerate (PHBV) is a biodegradable aliphatic polyester.

Chemistry Guide for Class 12 PSEB Polymers Textbook Questions and Answers

Question 1.
What are polymers?
Answer:
Polymers are high molecular mass macromolecules, which consist of repeating structural units derived from monomers. Polymers have a high molecular mass (103 – 107 u). In a polymer, various monomer units are joined by strong covalent bonds. These polymers can be natural as well as synthetic. Polythene, rubber, and nylon 6, 6 are examples of polymers.

Question 2.
How are polymers classified on the basis of structure?
Answer:
Polymers are classified on the basis of structure as follows:
1. Linear polymers: These polymers are formed of long straight chains. They can be depicted as :

e.g., high-density polythene (HDP), polyvinyl chloride PVC, etc.

2. Branched-chain polymers: These polymers are basically linear chain polymers with some branches. These polymers are represented as:

e.g., low-density polythene (LDP), amylopectin, etc.

3. Cross-linked or Network polymers: These polymers have many cross-linking bonds that give rise to a network-like structure. These polymers contain bi-functional and tri-functional monomers and strong covalent bonds between various linear polymer chains. Examples of such polymers include bakelite and Melmac.

Question 3.
Write the names of monomers of the following polymers:

Answer:
(i) Hexamethylenediamine [H₂N—(CH₂)₆—NH₂] and adipic acid [HOOC—(CH₂)₄ —COOH].
(ii) Caprolactam

(iii) Tetrafluoroethene, (F₂C – CF₂).

Question 4.
Classify the following as addition and condensation polymers: Terylene, Bakelite, Polyvinyl chloride, Polythene.
Answer:
Addition polymers: Polyvinyl chloride, polythene.
Condensation polymers: Terylene, bakelite.

Question 5.
Explain the difference between Buna-N and Buna-S.
Answer:
Both are copolymers. Buna-N is a copolymer of 1, 3-butadiene and acrylonitrile while Buna-S is a copolymer of 1,3-butadiene and styrene.

Question 6.
Arrange the following polymers in increasing order of their intermolecular forces:
(i) Nylon-6,6, Buna-S, Polythene
(ii) Nylon-6, Neoprene, Polyvinylchloride.
Answer:
(i) Buna-S < Polythene < Nylon-6,6
(ii) Neoprene < Polyvinyl chloride < Nylon-6

Punjab State Board PSEB 12th Class Chemistry Book Solutions Chapter 16 Chemistry in Everyday Life Textbook Exercise Questions and Answers.

PSEB Solutions for Class 12 Chemistry Chapter 16 Chemistry in Everyday Life

PSEB 12th Class Chemistry Guide Chemistry in Everyday Life InText Questions and Answers

Question 1.
Why do we need to classify drugs in different ways?
Answer:
Different ways of classification of drugs and the usefulness of such classification are as follows :

1. Classification on the basis of pharmacological effect is useful for doctors because it provides them the whole range of drugs available for the treatment of a particular type of problem.
2. Classification on the basis of drug action on a particular biochemical process is useful for choosing the correct lead compound for designing the synthesis of a desired drug.
3. Classification on the basis of molecular targets is useful for medicinal chemists so that they can design a drug which is most effective for a particular receptor site.
4. Classification on the basis of chemical structure is useful for doctors to design the synthesis of a number of structurally similar compounds having different substituents and then choosing the drug having the least toxicity.

Question 2.
Explain the term target molecules or drug targets as used in medicinal chemistry.
Answer:
Drugs interact with macromolecules such as proteins, carbohydrates, lipids, enzymes and nucleic acids. Hence, these are called drug targets. Drugs possessing some common structural features may have the same mechanism of action on targets.

Question 3.
Name the macro-molecules that are chosen as drug targets.
Answer:
Nucleic acids, proteins, carbohydrates, lipids, enzymes are chosen as drug targets.

Question 4.
Why should not medicines be taken without consulting doctors?
Answer:
Side effects are caused when a drug binds to more than one receptor site. So, a doctor must be consulted to choose the right drug which has the maximum affinity for a particular receptor site to have the desired effect. The dose of the drug is also crucial because some drugs like opiates in higher doses act as poisons and may cause death.

Question 5.
Define the term chemotherapy.
Answer:
The branch of chemistry which deals with the treatment of diseases using chemicals is called chemotherapy.

Question 6.
Which forces are involved in holding the drugs to the active site of enzymes?
Answer:
Ionic bonding, hydrogen bonding, van der Waals’ interaction, dipole-dipole interaction etc., are involved in holding the drugs to the active site of enzymes.

Question 7.
While antacids and antiallergic drugs interfere with the function of histamines, why do these not interfere with the function of each other?
Answer:
These (antacids and antiallergic drugs) do not interfere with the function of each other because they work on different receptors in the body.

Question 8.
Low level of noradrenaline is the cause of depression. What type of drugs are needed to cure this problem? Name two drugs.
Answer:
In event of low level of neurotransmitters, noradrenaline, antidepressant drugs are required. These drugs inhibit the enzymes which catalyse the degradation of noradrenaline. If the enzyme is inhibited, noradrenaline is slowly metabolised and thus activates its receptor for longer periods of time thereby reducing depression. Two important drugs are iproniazid and phenelzine.

Question 9.
What is meant by the term ‘broad-spectrum antibiotics? Explain.
Answer:
Broad-spectrum antibiotics are effective against several different types of harmful bacteria. Examples are tetracycline, ofloxacin, chloramphenicol, etc. Chloramphenicol can be used in case of typhoid, acute fever, dysentery, urinary infections, meningitis and pneumonia.

Question 10.
How do antiseptics differ from disinfectants? Give one example of each.
Answer:
Differences between antiseptics and disinfectants are as follows :
Antiseptics

• Antiseptics are chemical substances which prevent the growth of microorganisms and may even kill them but are not harmful to living tissues.
• Antiseptics are generally applied to living tissues such as wounds, cuts, ulcers and diseased skin surfaces.
• Dettol, furnace, soframicine are antiseptics.

Disinfectants

• Disinfectants are chemical substances which kill microorganisms or stop their growth but are harmful to human tissues.
• Disinfectants are applied to inanimate objects such as floor, drainage system, instruments, etc.
• Chlorine in the concentration of 0.2 to 0.4 ppm in aqueous solution and SO 2 in very low concentration are disinfectants.

Question 11.
Why are cimetidine and ranitidine better antacids than sodium hydrogen carbonate or magnesium or aluminium hydroxide?
Answer:
Excessive use of sodium hydrogen carbonate or a mixture of aluminium and magnesium hydroxide can make the stomach alkaline and trigger the production of even more acid. On the other hand, ranitidine and cimetidine prevent the interaction of histamine with the receptors present in the stomach wall. This results in release of lesser amount of acid. Thus, these are better antacids.

Question 12.
Name a substance which can be used as an antiseptic as well as disinfectant.
Answer:
0.2% solution of phenol acts as an antiseptic while 1% of the solution acts as a disinfectant.

Question 13.
What are the main constituents of Dettol?
Answer:
Chloroxylenol and a-terpineol in a suitable solvent.

Question 14.
What is tincture of iodine? What is its use?
Answer:
A 2-3 per cent solution of iodine in alcohol-water mixture is known as tincture of iodine. It is used as an antiseptic.

Question 15.
What are food preservatives?
Answer:
Chemical substances which are used to protect food against bacteria, yeasts and moulds are called food preservatives. For example, sodium metabisulphite, sodium benzoate, etc.

Question 16.
Why is the use of aspartame limited to cold foods and drinks?
Answer:
Aspartame decomposes on heating and may not work well. So, its use as an artificial sweetener is limited to foods and drinks at low temperatures.

Question 17.
What are artificial sweetening agents? Give two examples.
Answer:
Artificial sweetening agents are the chemical substances which provide sweetness to the food without increasing the calories to the body. For example, saccharin, aspartame, sucralose etc.

Question 18.
Name the sweetening agent used in the preparation of sweets for a diabetic patient.
Answer:
Saccharin.

Question 19.
What problem arises in using alitame as artificial sweetener?
Answer:
Alitame is a high potency artificial sweetener. Therefore, it becomes difficult to control die level of sweetness while using it.

Question 20.
How are synthetic detergents better than soaps?
Answer:
Advantages of synthetic detergents over soaps :

• Detergents can work with hard water too while soaps cannot.
• They can work even in an acidic medium while soaps cannot.
• Synthetic detergents are stronger cleansing agents than soaps.
• Their solubility is higher than that of soaps.
• They are prepared from hydrocarbons (petroleum) so their use is to save vegetable oils which are used during the preparation of soaps.

Question 21.
Explain the following terms with suitable examples :
(i) Cationic detergents
(ii) Anionic detergents and
(iii) Non-ionic detergents
Answer:
(i) Cationic detergents: These are quaternary ammonium salts of amines with acetates, chlorides or bromides.
Example: Cetyl trimethyl ammonium bromide
(ii) Anionic detergents: These detergents have large anionic part in their molecules. These are of two types :
(a) Sodium alkyl sulphates: For example, Sodium lauryl sulphate CH₃ (CH₂)₁₀ CH₂OSO₃^–Na⁺.
(b) Sodium alkyl benzene sulphonates: For example, sodium dodecylbenzene sulphonate

(iii) Non-ionic detergents: These are esters of high molecular mass alcohols with fatty acids. For example, Polyethylene glycol stearate.
CH₃ (CH₂)₁₆ COO(CH₂CH₂O)_nCH₂CH₂OH.

Question 22.
What are biodegradable and non-biodegradable detergents? Give one example of each.
Answer:
Detergents having straight hydrocarbon chains are easily degraded by microorganisms and hence are called biodegradable detergents, whereas detergents containing branched hydrocarbon chains are not easily degraded by the microorganisms and hence are called non-biodegradable detergents. As a result, non-biodegradable detergents accumulate in rivers and waterways thereby causing severe water pollution.

Examples of biodegradable detergents are : sodium lauryl sulphate, sodium 4-(l-dodecyl) benzene-sulphonate and sodium 4-(2-dodecyl) benzenesulphonate. An example of non-biodegradable detergent is sodium 4-(l, 3, 5, 7-tetramethyloctyl) benzenesulphonate.

Question 23.
Why do soaps not work with hard water?
Answer:
Calcium and magnesium salts present in hard water react with soaps to form insoluble compounds, which form curdy white precipitates and are difficult to remove from the clothes.

Question 24.
Can you use soaps and synthetic detergents to check the hardness of water?
Answer:
We can use soaps to check the hardness of water because with hard water, soaps give a gummy mass (sticky precipitate) but we cannot use detergents for this purpose because they give foam with both hard and soft water.

Question 25.
Explain the cleansing action of soaps.
Answer:
The cleansing action of soap is due to the fact that soap molecules, such as sodium stearate form micelle around the oil droplet in such a way that hydrophobic part of the stearate ions is in the oil droplet and hydrophilic part projects out of the oil droplet like the bristles.

Since the polar groups can interact with water, the oil droplet surrounded by stearate ions is now pulled in water and removed from the dirty surface. Thus, soap helps in emulsification and washing away of oils and fats. The negatively charged sheath around the globules prevents them from coming together and forming aggregates.

Question 26.
If the water contains dissolved calcium hydrogen carbonate, out of soaps and synthetic detergents which one will you use for cleaning clothes?
Answer:
Synthetic detergents.

Question 27.
Label the hydrophilic and hydrophobic parts in the following compounds.
(i) CH₃(CH₂)₁₀CH₂OSO₃^–Na⁺
(ii) CH₃(CH₂)₁₅N⁺(CH₃)₃ Br^–
(iii) CH₃(CH₂)₁₆COO(CH₂CH₂O)_n CH₂CH₂OH
Answer:

Chemistry Guide for Class 12 PSEB Chemistry in Everyday Life Textbook Questions and Answers

Question 1.
Sleeping pills are recommended by doctors to the patients suffering from sleeplessness but It is not advisable to take Its doses without consultation with the doctor. Why?
Answer:
Sleeping pills contain drugs that may be tranquillizers or antidepressants. They affect the nervous system, relieve anxiety, stress, irritability or excitement. But they should strictly be used under the supervision of a doctor. If not, the uncontrolled and overdosage can cause harm to the body and mind because in higher doses these drugs act as poisons.

Question 2.
With reference to which classification has the statement, “ranitidine is an antacid” been given?
Answer:
This statement refers to the classification according to the pharmacological effect of the drug because any drug which will be used to counteract the effect of excess acid in the stomach will be called antacid.

Question 3.
Why do we require artifical sweetening agents?
Answer:
Natural sweeteners (sucrose etc.) provide calories to the body. Taking extra calories is harmful for diabetic patients. So, artificial sweeteners are used (i) to control intake of calories and (ii) as a substitute of sugar for diabetics.

Question 4.
Write the chemical equation for preparing sodium soap from glyceryl oleate and glyceryl palmitate. Structural formulae of these compounds are given below.
(i) (C₁₅H₃₁COO)₃C₃H₅ – Glyceryl pfi]mitate
(ii) (C₁₇H₃₂COO)₃C₃H₅ – Glyceryl oleate
Answer:

Question 5.
Following types of non-ionic detergents are present in liquid detergents, emulsifying agents and wetting agents. Label the hydrophilic and hydrophobic parts In the molecule. Identify the functional group(s) present In the molecule.

Answer:

(b) Functional groups: Ether and alcohol.

Punjab State Board PSEB 12th Class Physics Book Solutions Chapter 8 Electromagnetic Waves Textbook Exercise Questions and Answers.

PSEB Solutions for Class 12 Physics Chapter 8 Electromagnetic Waves

PSEB 12th Class Physics Guide Electromagnetic Waves Textbook Questions and Answers

Question 1.
Figure 8.6 shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15 A.

(a) Calculate the capacitance and the rate of charge of potential difference between the plates.
(b) Obtain the displacement current across the plates.
(c) Is Kirchhoffs first rule (junction rule) valid at each plate of the capacitor? Explain.
Answer:
(a) Capacitance of capacitor is given by the relation
C = \(\frac{\varepsilon_{0} A}{d}\) = \(\frac{8.854 \times 10^{-12} \times \pi \times(0.12)^{2}}{5 \times 10^{-2}}\)
= 8.01F
Also \(\frac{d Q}{d t}\) = \(\frac{d V}{d t}\)
∴ \(\frac{d V}{d t}\) = \(\frac{0.15}{8.01 \times 10^{-12}}\)
= 1.87 × 10¹⁰V /s

(b) Displacement current I_d = ε₀ × \(\frac{d}{d t}\) (Φ_E)
Again Φ_E – EA across Hence,(negative end constant).
Hence, I_d = ε₀ A\(\frac{d E}{d t}\)
Again, E = \(\frac{Q}{\varepsilon_{0} A}\)
So, \(\frac{d E}{d t}=\frac{i}{\varepsilon_{0} A}\)
which corresponds i_d = i = 1.5A

(c) Yes, Kirchhoffs law is valid provided by current, we mean the sum of condition and displacement current.

Question 2.
A parallel plate capacitor (Fig. 8.7) made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad s^-1.

(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of B at a point 3.0 cm from the axis between the plates.
Answer:
(a) I_rms = V_rms × Cω
= 230 × 100 × 10¹² × 300
= 6.9 × 10^-6 A = 6.9 μ A

(b) Yes, we know that the deviation is correct even if I is steady DC or AC (oscillating in time) can be proved as
I_d = ε₀\(\frac{d}{d t}\) (σ) = ε₀\(\frac{d}{d t}\) (EA) (> σ = EA)
ε₀A \(\frac{d E}{d t}\) = ε₀A \(\frac{d}{d t}\) (\(\frac{\sigma}{\varepsilon_{0}}\))
ε₀A \(\frac{d}{d t}\) (\(\frac{\sigma}{\varepsilon_{0} A}\)) (> σ = \(\frac{q}{A}\))
ε₀A × \(\frac{1}{\varepsilon_{0} A} \cdot \frac{d q}{d t}\) = I
which is the required proof.

(c) The region formula for magnetic field
B = \(\frac{\mu_{0} r}{2 \pi R^{2}}\)i_d
even if I_d is oscillating (and so magnetic field B): The formula is valid. I_D oscillates in phase as i₀ = i (peak value of current). Now, we have
B₀ = \(\frac{\mu_{0} r}{2 \pi R^{2}}\)i₀
where B₀ and i₀ are the amplitude of magnetic field and current respectively.
So, i₀ = √2I_rms = 6.96 × 1.414 μA = 9.76μA
Given, r = 3 cm, R = 6cm
B₀ = \(\frac{\mu_{0} r i_{0}}{2 \pi R^{2}}\)
= \(\frac{10^{-7} \times 2 \times 3 \times 10^{-2} \times 9.76 \times 10^{-6}}{(6)^{2} \times\left(10^{-2}\right)^{2}}\)
= 1.633 × 10^-11 T

Question 3.
What physical quantity is the same for X-rays of wavelength 10^-10 m, red light of wavelength 6800 Å and radiowaves of wavelength 500 m?
Answer:
X-rays, red light and radiowaves all are the electromagnetic waves. They have different wavelengths and frequencies. But the physical quantity which is same for all of these is the velocity of light in vacuum which is denoted by c and is equal to 3 × 10⁸ ms^-1 W

Question 4.
A plane electromagnetic wave travels in vacuum along z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is 30 MHz, what is its wavelength?
Answer:
In an electromagnetic wave’s propagation vector \(\vec{K}\), electric field vector \(\vec{E}\) and magnetic field vector \(\vec{K}\) form a right handed system. As the propagation vector is along Z-direction, electric field vector will be along X-direction and magnetic field vector will be along Y-direction.
Frequency v = 30 MHz = 30 × 10⁶Hz
Speed of light c = 3 × 10⁸ ms^-1
Wavelength, λ = \(\frac{c}{v}\) = \(\frac{3 \times 10^{8}}{30 \times 10^{6}}\) = 10 m

Question 5.
A radio can tune in to any station in the 7.5 MHz to 12 MHz hand. What is the corresponding wavelength band?
Answer:
Speed of wave c = 3 × 10⁸ ms^-1
When frequency, V₁ = 7.5MHz = 7.5 × 10⁶ Hz
Wavelength, λ₁ = \(\frac{c}{v_{1}}\) = \(\frac{3 \times 10^{8}}{7.5 \times 10^{6}}\) = 40m
When frequency, V₂ 12 MHZ = 12 × 10⁶HZ
Wavelength, λ₂ = \(\frac{c}{v_{2}}\) = \(\frac{3 \times 10^{8}}{12 \times 10^{6}}\) = 25m
Wavelength band is from 25 m to 40 m.

Question 6.
A charged particle oscillates about its mean equilibrium position with a frequency of 10⁹ Hz. What is the frequency of the electromagnetic waves produced by the oscillator?
Answer:
According to Maxwell’s theory, an oscillating charged particle with a frequency v radiates electromagnetic waves of frequency v.
So, the frequency of electromagnetic waves produced by the oscillator is v = 10⁹ Hz.

Question 7.
The amplitude of the magnetic field part of a harmonic electromagnetic wave in vacuum is B₀ =510 nT. What is the amplitude of the electric field part of the wave?
The relation between magnitudes of magnetic and electric field vectors in vacuum is
\(\frac{E_{0}}{B_{0}}\) = c
⇒ E₀ = B₀C
Here, B₀ = 510 × 10^-9T, c = 3 × 10⁸ ms^-1
E₀ = 510 × 10^-9 × 3 × 10⁸ = 153N/C

Question 8.
Suppose that the electric field amplitude of an electromagnetic wave is E₀ = 120 N/C and that its frequency is v = 50.0 MHz. (a) Determine, B₀, ω, k and λ. (b) Find expressions for E and B.
Answer:
Electric field amplitude, E₀ = 120 N/C
Frequency of source, v = 50.0 MHz = 50 × 10⁶ Hz
Speed of light, c = 3 × 10⁸ m/s

(a) Magnitude of magnetic field strength is given as
B₀ \(\frac{E_{0}}{\mathcal{C}}\) = \(\frac{120}{3 \times 10^{8}}\)
40 × 10^-8T
= 400 × 10^-9 T
= 400 nT
Angular frequency of source is given as
ω = 2πv = 2π × 50 × 10⁶
= 3.14 × 10⁸ rad/s
Propagation constant is given as
k = \(\frac{\omega}{c}\) = \(\frac{3.14 \times 10^{8}}{3 \times 10^{8}}\) = 1.05 rad /m
Wavelength of wave is given us
λ = \(\frac{c}{v}\) = \(\frac{3 \times 10^{8}}{50 \times 10^{6}}\) = 6.0m

(b) Suppose the wave is propagating in the positive x direction. Then, the electric field vector will be in the positive y direction and the magnetic field vector will be in the positive z direction. This is because all three vectors are mutually perpendicular.
Equation of electric field vector is given as
\(\vec{E}\) = E₀sin (kx – ωt) ĵ
= 120 sin [1.05 x – 3.14 × 10⁸t] ĵ
And, magnetic field vector is given as
\(\vec{B}\) = B₀ sin (kx – ωt)k̂
\(\vec{B}\) = (4 × 10^-7)sin[1.05 x – 3.14 × 10⁸t]k̂

Question 9.
The terminology of different parts of the electromagnetic spectrum is given in the text. Use the formula E – hv (for energy of a quantum of radiation : photon) and obtain the photon energy in units of eV for different parts of the electromagnetic spectrum. In what way are the different scales of photon energies that you obtain related to the sources of electromagnetic radiation?
Answer:
Energy of a photon is given as
E = hv = \(\frac{h c}{\lambda}\)
where,
h = Planck’s constant = 6.6 × 10^-34 Js
c = Speed of light = 3 × 10⁸ m/s
λ = Wavelength of radiation
∴ E = \(\frac{6.6 \times 10^{-34} \times 3 \times 10^{8}}{\lambda}\) = \(\frac{19.8 \times 10^{-26}}{\lambda}\) = J
= \(\frac{19.8 \times 10^{-26}}{\lambda \times 1.6 \times 10^{-19}}\) = \(\frac{12.375 \times 10^{-7}}{\lambda}\) = eV
The given table lists the photon energies for different parts of an electromagnetic spectrum for different λ.

The photon energies for the different parts of the spectrum of a source indicate the spacing of the relevant energy levels of the source.

Question 10.
In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × 10¹⁰Hz and amplitude 48 Vm^-1.
(a) What is the wavelength of the wave?
(b) What is the amplitude of the oscillating magnetic field?
(c) Show that the average energy density of the E field equals the average energy density of the B field, [c = 3 × 10⁸ ms^-1]
Answer:
Frequency of the electromagnetic wave, v = 2.0 × 10¹⁰ Hz
Electric field amplitude, E₀ = 48 V m^-1
Speed of light, c = 3 × 10⁸ m/s

(a) Wavelength of the wave is given as
λ = \(\frac{\mathcal{C}}{\mathrm{v}}\) = \(\frac{3 \times 10^{8}}{2 \times 10^{10}}\) 0.015 m

(b) Magnetic field strength is given as
B₀ = \(\frac{E_{0}}{c}\)
= \(\frac{48}{3 \times 10^{8}}\) = 1.6 × 10^-7 T

(c) Let U_E and U_B be the energy density of \(\) field and \(\) field respectively. Energy density of the electric field is given as
U_E = \(\frac{1}{2}\) ε₀E²
And, energy density of the magnetic field is given as
U_B = \(\frac{1}{2 \mu_{0}}\)²
We have the relation connecting E and B as
E = cB ………….. (1)
where,
c = \(\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}}\) ……………. (2)
Putting equation (2) in equation (1), we get
E = \(\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}}\)B
Squaring both sides, we get
E² = \(\frac{1}{\varepsilon_{0} \mu_{0}}\) B²
ε₀E² = \(\frac{B^{2}}{\mu_{0}}\)
\(\frac{1}{2}\)ε₀E² = \(\frac{1}{2} \frac{B^{2}}{\mu_{0}}\)
⇒ U_E = E_B

Question 11.
Suppose that the electric field part of an electromagnetic wave in vacuum is
E = {(3.1 N/C) cos [(1.8 rad/m) y + (5.4 × 10⁶ rad/s) t]}î
(a) What is the direction of propagation?
(b) What is the wavelength λ ?
(c) What is the frequency v?
(d) What is the amplitude of the magnetic field part of the wave?
(e) Write an expression for the magnetic field part of the wave.
Answer:
(a) Wave is propagating along negative y-axis.

(b) Standard equation of wave is \(\vec{E}\) = E₀ cos(ky + cot)î
Comparing the given equation with standard equation, we have
E₀ = 3.1 N/C, k = 1.8 rad/m, ω = 5.4 × 10⁶ rad/s
Propagation constant k = \(\frac{2 \pi}{\lambda}\)
∴ λ = \(\frac{2 \pi}{k}\) = \(\frac{2 \times 3.14}{1.8}\) m = 3.49 m

(c) We have ω = 5.4 × 10⁶ rad/s
Frequency, v = \(\frac{\omega}{2 \pi}\) = \(\frac{5.4 \times 10^{6}}{2 \times 3.14}\) Hz
= 8.6 × 10⁵ Hz

(d) Amplitude of magnetic field,
B₀ = \(\frac{E_{0}}{c}\) = \(\frac{3.1}{3 \times 10^{8}}\) = 1.03 × 10^-8 T

(e) The magnetic field is vibrating along Z-axis because \(\vec{K}\),\(\vec{E}\),\(\vec{B}\) form a right handed system -ĵ × î = k̂
> Expression for magnetic field is
\(\vec{B}\) = B₀ cos(ky+ ωt)k̂
= [1.03 × 10^-8Tcos{(1.8rad / m) y +(5.4 × ⁶ rad/s)t}]k̂

Question 12.
About 5% of the power of a 100 W light bulb is converted to visible radiation. What is the average intensity of visible radiation
(a) at a distance of 1 m from the bulb?
(b) at a distance of 10 m?
Assume that the radiation is emitted isotropically and neglect reflection.
Answer:
Power in visible radiation, P = \(\frac{5}{100}\) × 100 = 5W
For a point source, intensity I = \(\frac{P}{4 \pi r^{2}}\), where r is distance from the source.

(a) When distance r = 1 m,
I = \(\frac{5}{4 \pi(1)^{2}}=\frac{5}{4 \times 3.14}\) = 0.4 W/m²

(b) When distance r = 10 m,
I = \(\frac{5}{4 \pi(10)^{2}}=\frac{5}{4 \times 3.14 \times 100}\)
= 0.004 W/m²

Question 13.
Use the formula λ_m T = 0.29 cm K to obtain the characteristic temperature ranges for different parts of the electromagnetic spectrum. What do the numbers that you obtain tell you?
Answer:
A body at a particular temperature produces a continuous spectrum of wavelengths. In case of a black body, the wavelength corresponding to maximum intensity of radiation is given according to Planck’s law. It can be given by the relation,
λ_m = \(\frac{0.29}{T}\) cm K
where, λ_m = maximum wavelength
T = temperature
Thus, the temperature for different wavelengths can be obtained as
For λ_m = 10^-4 cm; T = \(\frac{0.29}{10^{-4}}\) = 2900°K
For λ_m = 5 × 10^-5 cm; T = \(\frac{0.29}{5 \times 10^{-5}}\) = 5800°K
For λ_m = 10^-6 cm; T = \(\frac{0.29}{10^{-6}}\) = 290000 °K and so on.

The numbers obtained tell us that temperature ranges are required for obtaining radiations in different parts of an electromagnetic spectrum. As the wavelength decreases, the corresponding temperature increases.

Question 14.
Given below are some famous numbers associated with electromagnetic radiations in different contexts in physics. State the part of the electromagnetic spectrum to which each belongs.
(a) 21 cm (wavelength emitted by atomic hydrogen in interstellar space).

(b) 1057 MHz (frequency of radiation arising from two close energy levels in hydrogen; known as Lamb shift).

(c) 2.7 K (temperature associated with the isotropic radiation filling all space-thought to be a relic of the ‘big-bang’ origin of the universe).

(d) 5890 Å – 5896 Å (double lines of sodium).

(e) 14.4 keV [energy of a particular transition in ⁵⁷ Fe nucleus associated with a famous high resolution spectroscopic method (Mossbauer spectroscopy)].
Answer:
(a) 21 cm belongs to short wavelength end of radiowaves (or Hertizan waves).

(b) Wavelength, λ = \(\frac{c}{v}\) = \(\frac{3 \times 10^{8}}{1057 \times 10^{6}}\) = 0.28 m = 28 cm.
This also belongs to short wavelength end of radiowaves.

(c) From relation λ_mT = 0.29 × 10^-2 K,
λ_m = \(\frac{0.29 \times 10^{-2} \mathrm{~K}}{T}=\frac{0.29 \times 10^{2}}{2.7}\)
= 0.107 × 10^-2m= 0.107 cm.
This corresponds to microwaves.

(d) Wavelength doublet 5890Å – 5896Å belongs to the visible region. These are emitted by sodium vapour lamp.

(e) From relation, E = \(\frac{h c}{\lambda}\)
we have λ = \(\frac{h c}{E}\)
λ = \(\frac{6.63 \times 10^{-34} \times 3 \times 10^{8}}{14.4 \times 10^{3} \times 1.6 \times 10^{-19}} \mathrm{~m}\)
= 0.86 × 10^-10 m = 0.86 Å
It belongs to the X-ray region of electromagnetic spectrum.

Question 15.
Answer the following questions :
(a) Long distance radio broadcasts use short-wave bands. Why?

(b) It is necessary to use satellites for long distance TV transmission. Why?

(c) Optical and radiotelescopes are built on the ground but X-ray astronomy is possible only from satellites orbiting the earth Why?

(d) The small ozone layer on top of the stratosphere is crucial for human survival. Why?

(e) If the earth did not have an atmosphere, would its average surface temperature be higher or lower than what it is now?

(f) Some scientists have predicted that a global nuclear war on the earth would be followed by a severe ‘nuclear winter’ with a devastating effect on life on earth. What might be the basis of this prediction?
Answer:
(a) Long distance radio broadcasts use short-wave bands because only these bands can be refracted by the ionosphere.

(b) Yes, it is necessary to use satellites for long distance TV transmissions because television signals are of high frequencies and high energies. Thus, these signals are not reflected by the ionosphere. Hence, satellites are helpful in reflecting TV signals. Also, they help in long distance TV transmissions.

(c) With reference to X-ray astronomy, X-rays are absorbed by the atmosphere. However, visible and radiowaves can penetrate it. Hence, optical and radiotelescopes are built on the ground, while X-ray astronomy is possible only with the help of satellites orbiting the Earth.

(d) The small ozone layer on the top of the stratosphere is crucial for human survival because it absorbs harmful ultraviolet radiations present in sunlight and prevents it from reaching the Earth’s surface.

(e) In the absence of an atmosphere, there would be no greenhouse effect on the surface of the Earth. As a result, the temperature of the Earth would decrease rapidly, making it chilly and difficult for human survival.

(f) A global nuclear war on the surface of the Earth would have disastrous consequences. Post nuclear war, the Earth will experience severe winter as the war will produce clouds of smoke -that would cover maximum parts of the sky, thereby preventing solar light form reaching the atmosphere. Also, it will lead to the depletion of the ozone layer.

Punjab State Board PSEB 12th Class Chemistry Book Solutions Chapter 4 Chemical Kinetics Textbook Exercise Questions and Answers.

PSEB Solutions for Class 12 Chemistry Chapter 4 Chemical Kinetics

PSEB 12th Class Chemistry Guide Chemical Kinetics InText Questions and Answers

Question 1.
From the rate expression for the following reactions, determine their order of reaction and the dimensions of the rate constants.
(i) 3 NO(g) → N₂O(g) Rate = k[NO]²
(ii) H₂O₂ (aq) +3I^– (aq) + 2H⁺ → 2H₂O (l) + \(\mathbf{I}_{3}^{-}\)
Rate = k[H₂O₂] [I^–]
(iii) CH₃CHO(g) → CH₄(g) + CO(g) Rate = k [CH₃CHO]^3/2
(iv) C₂H₅Cl(g) → C₂H₄(g) + HCl (g) Rate = k [C₂H₅Cl]
Solution:
(i) Given, rate = k [NO]²
Therefore, order of the reaction = 2
Dimension of rate constant (k) = \(\frac{\text { Rate }}{[\mathrm{NO}]^{2}}\)
= \(\frac{m o l L^{-1} s^{-1}}{\left(m o l L^{-1}\right)^{2}}\)
= \(\frac{\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}}{\mathrm{~mol}^{2} \mathrm{~L}^{-2}}\)
= L mol^-1 s^-1

(ii) Given, rate = k [H₂O₂] [I^– ]
Therefore, order of the reaction = 2
Dimension of k = \(\frac{\text { Rate }}{\left[\mathrm{H}_{2} \mathrm{O}_{2}\right]\left[\mathrm{I}^{-}\right]}\)
= \(\frac{\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}}{\left(\mathrm{~mol} \mathrm{~L}^{-1}\right)\left(\mathrm{mol} \mathrm{L}^{-1}\right)}\)
= L mol^-1 s^-1

(iii) Given rate = k[CH₃CHO]^3/2
Therefore, order of the reaction = \(\frac{3}{2}\)
Dimension of k = \(\frac{\text { Rate }}{\left[\mathrm{CH}_{3} \mathrm{CHO}\right]^{3 / 2}}\)

= \(\frac{\text { mol L }^{-1} \mathrm{~s}^{-1}}{\mathrm{~mol}^{\frac{3}{2}} \mathrm{~L}^{-\frac{3}{2}}}\)
= mol ^-1/2L^1/2 s^-1

(iv) Given, rate = k [C₂H₅Cl]
Therefore, order of the reaction = 1
Dimension of k = \(\frac{\text { Rate }}{\left[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}\right]}\)
= \(\frac{\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}}{\mathrm{~mol} \mathrm{~L}^{-1}}\) = s^-1

Question 2.
For the reaction:
2A + B → A₂B
the rate = k[A] [B]² with k = 2.0 x 10^-6 mol^-2L²s^-1. Calculate the initial rate of the reaction when [A] = 0.1 mol L^-1, [B] = 0.2 mol L^-1. Calculate the rate of reaction after [A] is reduced to 0.06 mol L^-1.
Solution:
The initial rate of the reaction is
Rate = k [A][B]²
= (2.0 × 10^-6 mol^-2 L² s^-1) (0.1 mol L^-11) (0.2 mol L^-1 )²
= 8.0 × 10^-9 mol L^-1 s^-1
When [A] is reduced from 0.1 mol L^-1 to 0.06 molL^-1, the concentration of A reacted = (0.1 – 0.06) mol L^-1 = 0.04 mol L^-1 Therefore, concentration of B reacted
= \(\frac{1}{2}\) × 0.04 mol L^-1 = 0.02 mol L^-1
Then, concentration of B available, [B] = (0.2 -0.02) mol L^-1
= 0.18 mol L^-1
After [A] is reduced to 0.06 mol L^-1, the rate of the reaction is given by,
Rate = k [A][B]²
= (2.0 × 10^-6 mol^-2 L² s^-1) (0.06 mol L^-1) (0.18 mol L^-1)²
= 3.89 × 10^-9 mol L^-1 s^-1

Question 3.
The decomposition of NH₃ on platinum surface is zero order reaction. What are the rates of production of N₂ and H₂ if k = 2.5 x 10^-4 mol^-1 L s^-1?
Solution:
The decomposition of NH₃ on platinum surface is represented by the following equation

For zero order reaction, rate = k
∴ \(-\frac{1}{2} \frac{d\left[\mathrm{NH}_{3}\right]}{d t}=\frac{d\left[\mathrm{~N}_{2}\right]}{d t}=\frac{1}{3} \frac{d\left[\mathrm{H}_{2}\right]}{d t}\)
= 2.5 × 10^-4 mol L^-1 s^-1
Therefore, the rate of production of N₂
\(\frac{d\left[\mathrm{~N}_{2}\right]}{d t}\) = 2.5 × 10^-4 mol L^-1 s^-1
The rate of production of H₂
\(\frac{d\left[\mathrm{H}_{2}\right]}{d t}\) = 3 × 2.5 × 10^-4 mol L^-1 s^-1
= 7.5 × 10^-4 mol L^-1 s^-1

Question 4.
The decomposition of dimethyl ether leads to the formation of CH₄, H₂ and CO and the reaction rate is given by
Rate = k [CH₃OCH₃]^3/2
The rate of reaction is followed by increase in pressure in a closed vessel, so the rate can also be expressed in terms of the partial pressure of dimethyl ether, i.e.,
Rate = k(PCH_3OCH3 )^3/2
If the pressure is measured in bar and time in minutes, then what are the units of rate and rate constants?
Solution:
If the pressure is measured in bar and time in minutes, then
Unit of rate = bar min^-1
Rate = k(PCH_3OCH3 )^3/2
⇒ k = \(\frac{\text { Rate }}{\left(p_{\mathrm{CH}_{3} \mathrm{OCH}_{3}}\right)^{3 / 2}}\)
=

Question 5.
Mention the factors that affect the rate of a chemical reaction.
Answer:
The factors that affect the rate of a chemical reaction are as follows :
(i) Nature of reactants: Ionic substances react more rapidly than covalent compounds because ions produced after dissociation are immediately available for reaction.

(ii) Concentration of reactants: Rate of a chemical reaction is direcdy proportional to the concentration of reactants.

(iii) Temperature: Generally rate of a reaction increases on increasing the temperature.

(iv) Presence of catalyst: In presence of catalyst, the rate of reaction generally increase and the equilibrium state is attained quickly in reversible reactions.

(v) Surface area of the reactants: Rate of reaction increases with increase in surface area of the reactants. That is why powdered form of reactants is preffered than their granular form.

Question 6.
A reaction is second order with respect to a reactant. How is the rate of reaction affected if the concentration of the reactant is (i) doubled (ii) reduced to half?
Solution:
Let the concentration of the reactant be [A] = a
Rate of reaction, R = k [A]² = ka²
(i) If the concentration of the reactant is doubled, i.e. [A] = 2a, then the rate of the reaction would be
R’ = k (2a)²
= 4ka² = 4R
Therefore, the rate of the reaction would increase by 4 times.

Question 7.
What is the effect of temperature on the rate constant of a reaction? How can this effect of temperature on rate constant be represented quantitatively?
Answer:
The rate constant is nearly doubled with a rise in temperature by 10° for a chemical reaction.

The temperature effect on the rate constant can be represented quantitatively by Arrhenius equation, k = Ae^-E_a/RT

Where, k is the rate constant, A is the Arrhenius factor or the frequency factor, R is the gas constant, T is the temperature, and E_a is the energy of activation for the reaction.

Question 8.
In a pseudo first order hydrolysis of ester in water, the following results were obtained:

t/s	0	30	60	90
[Ester]/molL-1	0.55	0.31	0.17	0.085

(i) Calculate the average rate of reaction between the time interval 30 to 60 seconds.
(ii) Calculate the pseudo first order rate constant for the hydrolysis of ester.
Solution:
(i) Average rate of reaction between the time interval, 30 to 60 seconds
= \(\frac{d[\text { Ester }]}{d t}\)
= \(\frac{0.31-0.17}{60-30}\)
= \(\frac{0.14}{30}\)
= 4.67 × 10^-3 mol L^-1 s^-1

Question 9.
A reaction is first order in A and second order in B.
(i) Write the differential rate equation.
(ii) How is the rate affected on increasing the concentration of B three times?
(iii) How is the rate affected when the concentrations of both A and B are doubled?
Solution:
(i) The differential rate equation will be
– \(\frac{d[\mathrm{R}]}{d t}\) = k[A][B]²

(ii) If the concentration of B is increased three times, then
– \(\frac{d[\mathrm{R}]}{d t}\) = k[A][3B]²
= 9.k [A][B]²
Therefore, the rate of reaction will increase 9 times.

(iii) When the concentrations of both A and B are doubled,
– \(\frac{d[\mathrm{R}]}{d t}\) = k[2A][2B]²
= 8.k [A] [B]²
Therefore, the rate of reaction will increase 8 times.

Question 10.
In a reaction between A and B, the initial rate of reaction (r₀) was measured for different initial concentrations of A and B as given below:

A/mol L-1	0.20	0.20	0.40
B/mol L-1	0.30	0.10	0.05
r0/mol L-1 s-1	5.07 × 10-5	5.07 × 105	1.43 × 10-4

What is the order of the reaction with respect to A and B?
Solution:
Let the order of the reaction with respect to A be x and with respect to B be y.
Therefore
r₀ = k [A]^x [B]^y
5.07 × 10^-5 = k[0.20]^x [0.30]^y …………. (i)
5.07 × 10^-5 = k[0.20]^x [0.10]^y …………. (ii)
1.43 × 10^-4 = k[0.40]^x [0.05]^y ……….. (iii)

Hence, the order of the reaction with respect to A is 1.5 and with respect to B is 0.

Question 11.
The following results have been obtained during the kinetic studies of the reaction:
2A + B → C + D

Determine the rate law and the rate constant for the reaction.
Solution:
Let the order of the-reaction with respect to A be x and with respect to B be y.
Therefore, rate of the reaction is given by,
Rate = k [A]^x [B]^y According to the question,
6.0 × 10^-3; = k[0.1]^x [0.1]^y …………. (i)
7.2 × 10^-2 = k[0.3]^x [0.2]^y …………… (ii)
2.88 × 10^-1 = k[0.3]^x [0.4]^y ………….. (iii)
2.40 × 10^-2 = k[0.4]^x [0.1]^y …………… (iv)
Dividing equation (iv) by (i), we get

Question 12.
The reaction between A and B is first order with respect to A and zero order with respect to B. Fill in the blanks in the following table:

Solution:
The given reaction is of the first order with respect to A and of zero order with respect to B.
Therefore, the rate of the reaction is given by,
Rate = k[A]¹[B]⁰
⇒ Rate = fc[A]
From experiment I, we get
2.0 × 10^-2 molL^-1 min^-1 = k(0.1 molL^-1)
⇒ k = 0.2 min^-1

From experiment II, we get
4.0 × 10^-2 molL^-1 min^-1 = 0.2 min^-1 [A]
⇒ [A] = 0.2 mol L^-1

From experiment III, we get
Rate = 0.2 min^-1; × 0.4 mol L^-1
= 0.08 mol L^-1 min^-1

From experiment IV, we get
2.0 × 10^-2 molL^-1 min^-1 = 0.2 min^-1 [A]
⇒ [A] = 0.1 mol L^-1

Question 13.
Calculate the half-life of a first order reaction from their rate constants given below:
(i) 200 s^-1
(ii) 2 min^-1
(iii) 4 years^-1
Solution:
Half life period for first order reaction, t_1/2 = \(\)
(i) t_1/2 = \(\frac{0.693}{200 \mathrm{~s}^{-1}}\) = 0.347 × 10^-2 s
= 3.47 × 10^-3 s
(ii) t_1/2 = \(\frac{0.693}{2 \min ^{-1}}\) = 0.35 mm
(iii) t_1/2 = \(\frac{0.693}{4 \text { years }^{-1}}\)= 0.173 years 4 years-1

Question 14.
The half-life for radioactive decay of ¹⁴C is 5730 years. An archaeological artifact containing wood had only 80% of the ¹⁴C found in a living tree. Estimate the age of the sample.
Solution:
Decay constant (k) = \(\frac{0.693}{t_{1 / 2}}\)
\(\frac{0.693}{5730}\) = years ^-1
Radioactive decay follows first order kinetics
t = \(\frac{2.303}{k}\) = log\(\frac{[R]_{0}}{[R]}\)
= \(\frac{\frac{2.303}{0.693}}{5730}\) × log \(\frac{100}{80}\)
= 1845 years
Hence, the age of the sample is 1845 years.

Question 15.
The experimental data for decomposition of N₂0₅
[2N₂O₅ → 4NO₂ + O₂]
in gas phase at 318K are given below:

(i) Plot [N₂O₅] against t.
(ii) Find the half-life period for the reaction.
(iii) Draw a graph between log [N₂O₅ ] and t.
(iv) What is the rate law?
(v) Calculate the rate constant
(vi) Calculate the half-life period from k and compare it with (ii).
Solution:
(i) The plot of [N₂O₅] against time is given below:

(ii) Initial concentration of N₂O₅ = 1.63 x 10^-2 M
Half of this concentration = 0.815 x 10^-2 M
Time corresponding to this concentration = 1440 s
Hence t_1/2 = 1440 s

(iii) For graph between log[N₂O₅] and time, we first find the values of log[N₂O₅]

Time (s)	102 × [N2O5] mol L-1	log [N2O5]
0	1.63	-1.79
400	1.36	-1.87
800	1.14	-1.94
1200	0.93	-2.03
1600	0.78	-2.11
2000	0.64	-2.19
2400	0.53	-2.28
2800	0.43	-2.37
3200	0.35	-2.46

(iv) The given reaction is of the first order as the plot, log [N₂0₅] v/s t, is a straight line. Therefore, the rate law of the reaction is
Rate = k [N₂O₅]

(v) From the plot, log [N₂O₅] v/s t, we get
Slope = \(\frac{-2.46-(-1.79)}{3200-0}\)
= \(\frac{-0.67}{3200}\)
Again, slope of the line of the plot log [N₂O₅] v/s t is given by
– \(\frac{k}{2.303}\)
Therefore we get
\(-\frac{k}{2.303}=-\frac{0.67}{3200}\)
k = \(\frac{0.67 \times 2.303}{3200}\)
= 4.82 × 10^-4s^-1

(vi) Half-life period (t_1/2) = \(\)
= \(\frac{0.693}{4.82 \times 10^{-4} \mathrm{~s}^{-1}}\) = 1438 s
Half-life period (t_1/2) is calculated from the formula and slopes are approximately the same.

Question 16.
The rate constant for a first order reaction is 60 s^-1. How much time will it take to reduce the initial concentration of the reactant to its 1/16^th value?
Solution:
For first order reaction
t = \(\frac{2.303}{k}\) log \(\frac{1}{(a-x)}\) …………. (i)
Given (a – x) = \(\frac{1}{16}\); k= 60 s^-1
Placing the values in equation (i)
t = \(\frac{2.303}{60 \mathrm{~s}^{-1}}\) log \(\frac{a \times 16}{a}\)
= \(\frac{2.303}{60 \mathrm{~s}^{-1}}\) log16 \(\frac{2.303}{60 \mathrm{~s}^{-1}}\) × 4 log 2
= \(\frac{2.303}{60 \mathrm{~s}^{-1}}\) × 4 × 0.3010
= 4.6 × 10^-2s
Hence, the required time is 4.6 × 10^-2 s.

Question 17.
During nuclear explosion, one of the products is ⁹⁰Sr with half-life of 28.1 years. If 1μ g of ⁹⁰Sr was absorbed in the bones of a newly born baby instead of calcium, how much of it will remain after 10 years and 60 years if it is not lost metabolically.
Solution:
As radioactive disintegration follows first order kinetics,
∴ Decay constant of ⁹⁰Sr, k = \(\frac{0.693}{t_{1 / 2}}\)
= \(\frac{0.693}{28.1 \mathrm{y}}\) = 2.466 × 10^-2y^-1

To calculate the amount left after 10 years
[R]₀ = 1μg, t = 10 years, k = 2.466 × 10^-2y^-1,[R] =?
k = \(\frac{2.303}{t}\) log \(\frac{[R]_{0}}{[R]}\)
2.466 × 10^-2 = \(\frac{2.303}{10}\) log \(\frac{1}{[R]}\)
or log[R] = – 0.1071
or [Rl = Antilog \(\overline{1}\).8929 = 0.78 14 μg

To calculate the amount left after 60 years
2.466 × 10^-2 = \(\frac{2.303}{60}\) log \(\frac{1}{[R]}\)
or log[R] = – 0.6425
or [R] = Antilog \(\overline{1}\).3575 = 0.2278 μg

Question 18.
For a first order reaction, show that time required for 99% completion is twice the time required for the completion of 90% of reaction.
Solution:
For a first order reaction, the time required for 99% completion is

Therefore, t₁ = 2t₂
Hence, the time required for 99% completion of a first order reaction is twice the time required for the completion of 90% of the reaction.

Question 19.
A first order reaction takes 40 min for 30% decomposition.
Calculate t_1/2
Solution:
Given, t = 40 min,
For a first order reaction,

Question 20.
For the decomposition of azoisopropane to hexane and nitrogen at 543 K, the following data are obtained.

t (sec)	P(mm of Hg)
0	35.0
360	54.0
720	63.0

Calculate the rate constant.
Solution:
The decomposition of azoisopropane to hexane and nitrogen at 543 K is represented by the following equation:

Hence, the average value of rate constant
k = \(\frac{\left(2.175 \times 10^{-3}\right)+\left(2.235 \times 10^{-3}\right)}{2} s^{-1}\)
= 2.21 × 10^-3s^-1

Question 21.
The following data were obtained during the first order thermal decomposition of SO₂Cl₂ at a constant volume.
SO₂Cl₂(g) → SO₂(g) + Cl₂(g)

Experiment	Time/s-1	Total pressure/atm
1	0	0.5
2	100	0.6

Calculate the rate of the reaction when total pressure is 0.65 atm.
Solution:
The first order thermal decomposition of SO₂cl₂ at a constant volume is represented by the following equation:

= 2.23 × 10^-3s^-1

When P_t = 0.65 atm,
P₀ + p = 0.65
⇒ p = 0.65 – P₀
= 0.65 – 0.5
= 0.15 atm
Pressure of SO₂Cl₂ at time t (P_SO2Cl2 SO₂Cl₂
= P₀ – P
= 0.5 – 0.15
= 0.35 atm

Therefore, the rate of equation, when total pressure is 0.65 atm, is given by,
Rate = k × (P_SO2Cl2 SO₂Cl₂)
= (2.23 × 10^-3 s^-1) (0.35 atm)
= 7.8 × 10^-5 atm s^-1

Question 22.
The rate constant for the decomposition of N₂O₅ at various temperatures is given below:

Draw a graph between In k and 1/T and calculate the values of A and E_a.
Predict the rate constant at 30° and 50°C.
Solution:
To draw the plot of log k versus 1/T, we can rewrite the given data as follows:

From graph, we find
Slope = \(\frac{-2.4}{0.00047}\) = 5106.38
E_a = – Slope × 2.303 × R
= – (- 5106.38) × 2.303 × 8.314
= 97772.58 J mol^-1
= 97.77258 kJ mol^-1

We know that,
log k = log A – \(\frac{E_{a}}{2.303 R T}\)
log k = log \(\left[-\frac{E_{a}}{2.303 R}\right] \frac{1}{T}\) = log A
Compare it with y = mx + c (which is equation of line in intercept form)
log A = value of intercept on y-axis i.e.
on log k-axis [y₂ – y₁ = -1 – (-7.2)]
= (-1 + 7.2) = 6.2 ,
log A = 6.2
A = Antilog 6.2
= 1.585 × 10⁶ s^-1
The values of rate constant k can be found from graph as follows:

We can also calculate the value of A from the following formula
log k = log A = \(\frac{E_{a}}{2.303 R T}\)

Question 23.
The rate constant for the decomposition of hydrocarbons is 2.418 × 10^-5 s^-1 at 546 K. If the energy of activation is 179.9 kJ/mol, what will be the value of pre-exponential factor.
Solution:
Given, k = 2.418 × 10^-5s^-1, T = 546 K
E_a = 179.9 kJ mol^-1 = 179.9 × 10³ J mol^-1
According to the Arrhenius equation,
k = Ae^-Ea/RT
ln k = ln A – \(\frac{E_{a}}{R T}\)
log k = log A – \(\frac{E_{a}}{2.303 R T}\)
log A = log K + \(\frac{E_{a}}{2.303 R T}\)
= log(2.418 × 10¹⁵s^-1) + \(\frac{179.9 \times 10^{3} \mathrm{~J} \mathrm{~mol}^{-1}}{2.303 \times 8.314 \mathrm{Jk}^{-1} \mathrm{~mol}^{-1} \times 546 \mathrm{~K}}\)
= (0.3835 – 5) +17.2082 = 12.5917
Therefore, A = antilog (12.5917) = 3.9 × 10¹²s^-1

Question 24.
Consider a certain reaction A → Products with k = 2.0 × 10^-2s^-1 Calculate the concentration of A remaining after 100 s if the initial concentration of A is 1.0 mol L^-1.
Solution:
Given, k = 2.0 x 10^-2s^-1, t = 100 s, [A]₀ = 1.0 mol L^-1
Since, the unit of k is s^-1, the given reaction is a first order reaction.

Question 25.
Sucrose decomposes in acid solution into glucose and fructose according to the first order rate law, with t_1/2 =3.00 hours. What fraction of sample of sucrose remains after 8 hours?
Solution:
For a first order reaction,

= 0.158 M
Hence, the fraction of sample of sucrose that remains after 8 hours is 0.158 M.

Question 26.
The decomposition of hydrocarbon follows the equation k = (45 × 10¹¹ s¹)e^-28000k/T
Calculate E_a.
Solution:
The given equation is
k = (45 × 10¹¹ s¹)e^-28000k/T …(i)
Arrhenius equation is given by,
k = Ae^E_a/RT …(ii)
From equation (i) and (ii), we get
\(\frac{E_{a}}{R T}\) = \(\frac{28000 \mathrm{~K}}{T}\)
⇒ E_a = R × 28000 K
= 8.314 J K^-1 mol^-1 × 28000 K
= 232792 J mol^-1
= 232.792 kJ mol^-1

Question 27.
The rate constant for the first order decomposition of H₂O₂ is given by the following equation :
log k = 14.34 – 1.25 × 10⁴ K/T
Calculate E_a for this reaction and at what temperature will its half-period be 256 minutes?
Solution:
Arrhenius equation is given by,
k = Ae^-E_a/RT
⇒ log k = log A – \(\frac{E_{a}}{2.303 \mathrm{RT}}\) …(i)
log k = 14.34 – 1.25 × 10⁴ K/T …(ii)
From equation (i) and (ii), we get
\(\frac{E_{a}}{2.303 \mathrm{RT}}\) = \(\frac{1.25 \times 10^{4} \mathrm{~K}}{T}\)
⇒ E_a = 1.25 × 10⁴K × 2.303 × R
= 1.25 × 10⁴K × 2.303 × 8.314 J K^-1 mol^-1
= 239339.3 J mol^-1
= 239.34 kJ mol^-1
Also, when t_1/2 = 256 minutes,
For first order reaction
k = \(\frac{0.693}{t_{1 / 2}}\)
= \(\frac{0.693}{256}\)
= 2.707 × 10^-13 min^-1
= 4.51 × 10^-5 s^-1
According to Arrhenius theory,
log k = 14.34 – 1.25 × 10^,4K/T

Question 28.
The decomposition of A into product has value of & as 45 × 10³ s^-1 at 10°C and energy of activation 60 kJ mol^-1. At what temperature would k be 1.5 × 10⁴ s^-1.
Solution:
From Arrhenius equation, we get
\(\log \frac{k_{2}}{k_{1}}\) = \(\frac{E_{a}}{2.303 \mathrm{R}}\left(\frac{T_{2}-T_{1}}{T_{1} T_{2}}\right)\)
Also, k₁ = 4.5 × 10³ s^-1
T₁ = 273 + 10 = 283k
k₂ = 1.5 × 10⁴ s^-1
E_a = 60 kJmol^-1 = 6.0 × 10⁴ Jmol^-1

⇒ 0.0472T₂ = T₂ – 283
⇒ 0.9528T₂ = 283
⇒ T₂ = 297.019 K
= 297K = (297 – 273)⁰C
= 24⁰C
Hence, k would be 1.5 × 10⁴ s^-1 at 24⁰C.

Question 29.
The time required for 10% completion of a first order reaction at 298 K is equal to that required for its 25% completion at 308 K. If the value of A is 4 × 10¹⁰ s^-1. Calculate k at 318 K and E_a.
Solution:
For a first order reaction,

To calculate k at 318 K,
It is given that, A = 4 × 10¹⁰ s^-1, T = 318 K
Again, from Arrhenius equation, we get
log k = log A – \(\frac{E_{a}}{2.303 \mathrm{RT}}\)
= log (4 × 10¹⁰) – \(\frac{76.64 \times 10^{3}}{2.303 \times 8.314 \times 318}\)
= (0.6021 + 10) – 12.5870 = -1.9849 k
k = Antilog (-1.9849)
= Antilog (2.0151) = 1.035 × 10^-2s^-1
E_a = 76.640 kJ mol^-1
E_a = 76.640 kJmol^-1
k = 1.035 × 10^-2s^-1

Question 30.
The rate of a reaction quadruples when the temperature changes from 293 K to 313 K. Calculate the energy of activation of the reaction assuming that it does not change with temperature.
Solution:
Given, k₂ = 4k₁, T₁ = 293 K, T₂ = 313 K
From Arrhenius equation, we get

Hence, the required energy of activation is 52.86 kJ mol^-1

Chemistry Guide for Class 12 PSEB Chemical Kinetics Textbook Questions and Answers

Question 1.
For the reaction R → P, the concentration of a reactant changes from 0.03 M to 0.02 M in 25 minutes. Calculate the average rate of reaction using units of time both in minutes and seconds.
Solution:

Question 2.
In a reaction, 2A → Products, the concentration of A decreases from 0.5 mol L^-1 to 0.4 mol L^-1 in 10 minutes. Calculate the rate during this interval?
Solution:
Rate of reaction = Rate of disappearance of A = – \(\frac{1}{2} \frac{\Delta[A]}{\Delta t}\)
= – \(\frac{1}{2} \frac{[A]_{2}-[A]_{1}}{t_{2}-t_{1}}\)
= – \(\frac{1}{2} \frac{(0.4-0.5) \mathrm{mol} \mathrm{L}^{-1}}{10 \mathrm{~min}}\)
= – \(\frac{1}{2} \frac{-0.1}{10}\)
= 0.005 mol L^-1 min^-1
= 5 × 10^-3 M min^-1

Question 3.
For a reaction, A + B → Product; the rate law is given by,
r = k [A]^1/2 [B]². What is the order of the reaction?
Solution:
The order of the reaction = \(\frac{1}{2}\) + 2
= 2\(\frac{1}{2}\) = 2.5

Question 4.
The conversion of molecules X to Y follows second order kinetics. If concentration of X is increased to three times how will it affect the rate of formation of Y? ‘
Solution:
The reaction X → Y follows second order kinetics.
Therefore, the rate equation for this reaction will be:
Rate (r) = k[X]² = k × X² …………. (i)
If the concentration of X is increased to three times, then
Rate (r’) = fc(3X)² = k × 9X² ………….. (ii)
Dividing eq. (ii) by eq. (i)
\(\frac{r^{\prime}}{r}=\frac{k \times 9 X^{2}}{k \times X^{2}}\) = 9
It means that the rate of formation of Y will increase by nine times.

Question 5.
A first order reaction has a rate constant 1.15 × 10^-3s^-1. How long will 5 g of this reactant take to reduce to 3 g?
Solution:
Initial amount [R]₀ = 5 g
Final amount [R] = 3 g
Rate constant (k) = 1.15 × 10^-3s^-1
We know that for a 1^st order reaction,

= 444.38 s
= 444 s

Question 6.
Time required to decompose SO₂Cl₂ to half of its initial amount is 60 minutes. If the decomposition is a first order reaction, calculate the rate constant of the reaction.
Solution:
We know that for a 1^st order reaction,
t_1/2 = \(\frac{0.693}{k}\)
> k = \(\frac{0.693}{t_{1 / 2}}\)
= \(\frac{0.693}{60 \mathrm{~min}}\) = \(\frac{0.693}{(60 \times 60) \mathrm{s}}\)
or k = 1.925 × 10^-4 s^-1]

Question 7.
What will be the effect of temperature on rate constant?
Answer:
The rate constant of a reaction is nearly doubled with a 10° rise in temperature. However, the exact dependence of the rate of a chemical reaction on temperature is given by Arrhenius equation,
k = Ae^-Ea/RT
Where, A is the Arrhenius factor or the frequency factor, T is the temperature, R is the gas constant, E_a is the activation energy.

Question 8.
The rate of the chemical reaction doubles for an increase of 10 K in absolute temperature from 298 K. Calculate E_a.
Solution:
Given, T₁ = 298 K
∴ T₂ = (298 + 10)K = 308K
We also know that the rate of the reaction doubles when temperature is increased by 10°.
Therefore, let us take the value of k₁ = k and that of k₂ = 2k
Also, R =8.314 JK^-1 mol^-1
Now, substituting these values in the equation:

Question 9.
The activation energy for the reaction
2HI (g) → H₂ + I₂(g)
is 209.5 kJ mol^-1 at 58IK. Calculate the fraction of molecules of reactants having energy equal to or greater than activation energy?
Solution:
Fraction of molecules of reactants (x) having energy equal to or greater than activation energy may be calculated as follows
or log x = \(\frac{-E_{a}}{R T}\) or log x = –\(\frac{E_{a}}{2.303 R T}\)
or log x = – \(\frac{209.5 \times 10^{3}}{2.303 \times 8.314 \times 581}\)
= -18.8323
x = Antilog (-18.8323) = Antilog (\(\overline{19}\).1677)
= 1.471 × 10^-19
Hence, fraction of molecules of reactants having energy equal to or greater than activation energy = 1.471 × 10^-19

Punjab State Board PSEB 12th Class Physics Book Solutions Chapter 7 Alternating Current Textbook Exercise Questions and Answers.

PSEB Solutions for Class 12 Physics Chapter 7 Alternating Current

PSEB 12th Class Physics Guide Alternating Current Textbook Questions and Answers

Question 1.
A 100 Ω resistor is connected to a 220 V, 50 Hz ac supply.
(a) What is the rms value of current in the circuit?
(b) What is the net power consumed over a full cycle?
Answer:
The given voltage of 220 V is the rms or effective voltage.
Given V_rms = 220 V, v = 50 Hz, R = 100 Ω
(a) RMS value of current,
I_rms = \(\frac{V_{r m s}}{R}\) = \(\frac{220}{100}\) = 2.2 A
Net power consumed, P = I²_rmsR
= (2.20)² × 100 = 484 W

Question 2.
(a) The peak voltage of an ac supply is 300 V. What is the rms voltage?
(b) The rms value of current in an ac circuit is 10 A. What is the peak current?
Answer:
(a) Given, V₀ = 300 V
V_rms = \(\frac{V_{0}}{\sqrt{2}}=\frac{300}{\sqrt{2}}\) = 150√2 ≈ 212 V

(b) Given, I_rms = 10 A
I₀ = I_rms √2 = 10 × 1.41 = 14.1 A

Question 3.
A 44 mH inductor is connected to 220 V, 50 Hz ac supply. Determine the rms value of the current in the circuit.
Answer:
Inductance of inductor, L = 44 mH = 44 × 10^-3 H
Supply voltage, V = 220 V
Frequency, v = 50 Hz
Angular frequency, ω = 2 πv
Inductive reactance, X_L = ωL = 2πvL × 2π × 50 × 44 × 10^-3Ω
rms value of current is given as
I = \(\frac{V}{X_{L}}\) = \(\frac{220}{2 \pi \times 50 \times 44 \times 10^{-3}}\) = 15.92 A
Hence, the rms value of current in the circuit is 15.92 A.

Question 4.
A 60 μF capacitor is connected to a 110 V, 60 Hz ac supply. Determine the rms value of the current in the circuit.
Answer:
Capacitance of capacitor, C = 60μF = 60 × 10^-6F
Supply voltage, V = 110 V
Frequency, v = 60 Hz
Angular frequency, ω = 2 πv
Capacitive reactance,
X_C = \(\frac{1}{\omega C}\) = \(\frac{1}{2 \pi v C}\) = \(\frac{1}{2 \pi \times 60 \times 60 \times 10^{-6}}\)Ω
rms value of current is given as
I = \(\frac{V}{X_{C}}\) = \(\frac{110}{\frac{1}{2 \pi \times 60 \times 60 \times 10^{-6}}}\)
= 110 × 2 × 3.14 × 3600 × 10^-6
= 2.49 A
Hence, the rms value of current in the circuit is 2.49 A.

Question 5.
In Exercises 7.3 and 7.4, what is the net power absorbed by each circuit over a complete cycle. Explain your answer.
Answer:
In the inductive circuit,
rms value of current, I = 15.92 A
rms value of voltage, V = 220 V
Hence, the net power absorbed by the circuit, can be obtained by the relation,
P = VIcosΦ
where,
Φ = Phase difference between V and I.
For a pure inductive circuit, the phase difference between alternating voltage and current is 90°i. e., Φ = 90°
Hence, P = 0 i. e., the net power is zero.

In the capacitive circuit,
rms value of current, I = 2.49 A
rms value of voltage, V = 110 V
Hence, the net power absorbed by the circuit, can be obtained as
P = VIcosΦ
For a pure capacitive circuit, the phase difference between alternating voltage and current is 90°i. e., Φ = 90 °
Hence, P = 0 i. e., the net power is zero.

Question 6.
Obtain the resonant frequency ω_r of a series LCR circuit with L = 2.0 H, C = 32 μF and R = 10 Ω. What is the Q-value of this circuit?
Answer:
Resonant frequency,
ω_r = \(\frac{1}{\sqrt{L C}}\) = \(\frac{1}{\sqrt{2.0 \times 32 \times 10^{-6}}}\)
= \(\frac{1}{8}\) × 10³ = 125 rads^-1
Q = \(\frac{\omega_{r} L}{R}\) = \(\frac{125 \times 2.0}{10}\) = 25

Question 7.
A charged 30 μF capacitor is connected to a 27 mH inductor.
What is the angular frequency of free oscillations of the circuit?
Answer:
Capacitance of the capacitor, C = 30 μF = 30 × 10^-6 F,
Inductance of the inductor, L = 27 mH = 27 × 10^-3H
Angular frequency is given as
ω_r = \(\frac{1}{\sqrt{L C}}\)
= \(\frac{1}{\sqrt{27 \times 10^{-3} \times 30 \times 10^{-6}}}\)
= \(\frac{1}{9 \times 10^{-4}}=\frac{10^{4}}{9}\)
= 1.11 × 10³ rad/s
Hence, the angular frequency of free oscillations of the circuit is 1.11 × 10³ rad/s.

Question 8.
Suppose the initial charge on the capacitor in Exercise 7.7 is 6 mC. What is the total energy stored in the circuit initially? What is the total energy at later time?
Answer:
Capacitance of the capacitor, C = 30 μF = 30 × 10^-6F
Inductance of the inductor, L = 27 mH = 27 × 10^-3 H
Charge on the capacitor, Q = 6 mC = 6 × 10^-3 C
Total energy stored in the capacitor can be calculated as
E = \(\frac{1}{2} \frac{Q^{2}}{C}\) = \(\frac{1}{2} \frac{\left(6 \times 10^{-3}\right)^{2}}{\left(30 \times 10^{-6}\right)}\)
= \(\frac{36 \times 10^{-6}}{2\left(30 \times 10^{-6}\right)}\)
= \(\frac{6}{10}\) = 0.6 J
Total energy at a later time will remain the same because energy is shared between the capacitor and the inductor.

Question 9.
A series LCR circuit with R = 20 Ω, L = 1.5 H and C = 35 μF is connected to a variable frequency 200 V ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?
Answer:
When frequency of supply is equal to natural frequency of circuit, then resonance is obtained. At resonance X_C = X_L
⇒ Impedance, Z = \(\sqrt{R^{2}+\left(X_{C}-X_{L}\right)^{2}}\)
= R = 20Ω
Current in circuit,
I_rms = \(\frac{V_{r m s}}{R}\) = \(\frac{200}{20}\) = 10A
Power factor
cosΦ = \(\frac{R}{Z}=\frac{R}{R}\) = 1
∴ Average power p_av = V_rms I_rms cosΦ = V_rms I_rms
= 20 × 10 = 2000 W = 2 kW

Question 10.
A radio can tune over the frequency range of a portion of MW broadcast band : (800 kHz to 1200 kHz). If its LC circuit has an effective inductance of 200 μH, what must be the range of its variable capacitor?
[Hint: For timing, the natural frequency i. e., the frequency of free oscillations of the LC circuit should be equal to the frequency of the radiowave.]
Answer:
The range of frequency (v) of the radio is 800 kHz to 1200 kHz
Lower tuning frequency, v₁ = 800 kHz = 800 × 10³ Hz
Upper tuning frequency, v₂ = 1200 kHz = 1200 × 10⁶ Hz
Effective inductance of circuit, L = 200 μH = 200 × 10^-6 H
Capacitance of variable capacitor for v₁ is given as
C₁ = \(\frac{1}{\omega_{1}^{2} L}\)
where, ω₁ = Angular frequency for capacitor C₁
= 2 πv₁
= 2 π × 800 × 10³ rad/s
∴ C₁ = \(\frac{1}{\left(2 \pi \times 800 \times 10^{3}\right)^{2} \times 200 \times 10^{-6}}\)
= 197.8 × 10^-12F
= 197.8 pF
Capacitance of variable capacitor for v₂ is given as
C₂ = \(\frac{1}{\omega_{2}^{2} L}\)
where,
ω₂ = Angular frequency for capacitor C₂
= 2πv₂
= 2 π × 1200 × 10³ rad/s
∴ C ₂ = \(\frac{1}{\left(2 \pi \times 1200 \times 10^{3}\right)^{2} \times 200 \times 10^{-6}}\)
= 87.95 × 10^-12 F = 87.95 pF
Hence, the range of the variable capacitor is from 87.95 pF to 197.8 pF.

Question 11.
Figure 7.21 shows a series LCR circuit connected to a variable frequency 230 V source. Z, = 5.0H, C = 80 μF, R = 40Ω.

(a) Determine the source frequency which drives the circuit in resonance.
(b) Obtain the impedance of the circuit and the amplitude of current at the resonating frequency.
(c) Determine the rms potential drops across the three elements of the circuit. Show that the potential drop across the LC combination is zero at the resonating frequency.
Answer:
Given, the rms value of voltage V_rms = 230 V
Inductance L = 5H
Capacitance C = 80 μF = 80 × 10^-6 F
Resistance R = 40 Ω

(a) For resonance frequency of circuit
ω_r = \(\frac{1}{\sqrt{L C}}\) = \(\frac{1}{\sqrt{5 \times 80 \times 10^{-6}}}\) = 50 rad/s
Source frequency at resonance, then
v₀ = \(\frac{\omega_{0}}{2 \pi}\) = \(\frac{50}{2 \times 3.14}\) = 7.76 Hz

(b) At the resonant frequency, X_L = X_C
So, impedance of the circuit Z = R
∴ Impedance Z = 40 Ω
The rms value of current in the circuit
I_rms = \(\frac{V_{r m s}}{Z}\) = \(\frac{230}{40}\) = 5.75 A
Amplitude of current, I₀ = I_rms √2
= 5.75 × √2 = 8.13 A

(c) The rms potential drop across I,
V_L = I_rms × X_L = I_rms × ω_rL
= 5.75 × 50 × 5 = 1437.5V
The rms potential drop across R
V_R = I_rms R = 5.75 × 40 = 230 V
The rms potential drop across C,
V_C = I_rms × X_C = I_rms × \(\frac{1}{\omega_{r} C}\)
= 5.75 × \(\frac{1}{50 \times 80 \times 10^{-6}}\)
= 1437.5V
Potential drop across LC combinations
= I_rms(X_L – X_C)
= I_rms (X_L – X_L) = 0
(∵ X_L = X_C in resonance)

Question 12.
An LC circuit contains a 20 mH inductor and a 50 μF capacitor with an initial charge of 10 mC. The resistance of the circuit is negligible. Let the instant the circuit is closed be t = 0.
(a) What is the total energy stored initially? Is it conserved during LC oscillations?
(b) What is the natural frequency of the circuit?
(c) At what time is the energy stored (i) completely electrical (Lestored in the capacitor)? (ii) completely magnetic (i.e., stored in the inductor)?
(d) At what times is the total energy shared equally between the inductor and the capacitor?
(e) If a resistor is inserted in the circuit, how much energy is eventually dissipated as heat?
Answer:
Inductance of the inductor, L = 20 mH = 20 × 10^-3H
Capacitance of the capacitor, C = 50 μF = 50 × 10^-6 F
Initial charge on the capacitor, Q = 10 mC = 10 × 10^-3C

(a) Total energy stored initially in the circuit is given as
E = \(\frac{1}{2} \frac{Q^{2}}{C}\)
= \(\frac{\left(10 \times 10^{-3}\right)^{2}}{2 \times 50 \times 10^{-6}}=\frac{10^{-4}}{10^{-4}}\) = 1J
Hence, the total energy stored in the LC circuit will be conserved because there is no resistor connected in the circuit.

(b) Natural frequency of the circuit is given by the relation,
v = \(\frac{1}{2 \pi \sqrt{L C}}\)
= \(\frac{1}{2 \pi \sqrt{20 \times 10^{-3} \times 50 \times 10^{-6}}}\)
= \(\frac{10^{3}}{2 \pi}\) = 159.24 Hz
Natural angular frequency,
ω_c = \(\frac{1}{\sqrt{L C}}\) = \(\frac{1}{\sqrt{20 \times 10^{-3} \times 50 \times 10^{-6}}}\)
= \(\frac{1}{\sqrt{10^{-6}}}\) = 10³ rad/s
Hence, the natural frequency of the circuit is 10 rad/s.

(c) (i) For time period (T = \(\frac{1}{v}\) = \(\frac{1}{159.24}\) = 6.28 ms), total charge on the
capacitor at time t,
Q’ = Q cos\(\frac{2 \pi}{T}\)t
For energy stored is electrical, we can write Q’ = Q
Hence, it can be inferred that the energy stored in the capacitor is completely electrical at time, t = 0, \(\frac{T}{2}\), T, \(\frac{3 T}{2}\),…

(ii) Magnetic energy is the maximum when electrical energy, Q’ is equal to 0.
Hence, it can be inferred that the energy stored in the capacitor is
completely magnetic at time, t = \(\frac{T}{4}\), \(\frac{3 T}{4}\), \(\frac{5 T}{4}\),….

(d) Q’ = Charge on the capacitor when total energy is equally shared between the capacitor and the inductor at time t.
When total energy is equally shared between the inductor and capacitor,
the energy stored in the capacitor = \(\frac{1}{2}\) (maximum energy)

Hence, total energy is equally shared between the inductor and the capacitor at time,
t = \(\frac{T}{8}\), \(\frac{3 T}{8}\),\(\frac{5 T}{8}\)

(e) If a resistor is inserted in the circuit, then total initial energy is dissipated as heat energy in the circuit. The resistance damps out the LC oscillation.

Question 13.
A coil of inductance 0.50 H and resistance 100 Ω is connected to a 240 V, 50 Hz ac supply.
(a) What is the maximum current in the coil?
(b) What is the time lag between the voltage maximum and the current maximum?
Answer:
Given, L = 0.50 H ,R = 100 Ω, V = 240 V, v = 50 Hz
(a) Maximum (or peak) voltage V₀ = V – √2
Maximum current, I₀ = \(\frac{V_{0}}{Z}\)
Inductive reactance, X_L = ω_L = 2πvL
= 2 × 3.14 × 50 × 0.50
= 157 Ω.
Z = \(\sqrt{R^{2}+X_{L}^{2}}\)
= \(\sqrt{(100)^{2}+(157)^{2}}\) = 186 Ω

Question 14.
Obtain the answers (a) to (b) in Exercise 7.13 if the circuit is connected to a high frequency supply (240 V, 10 kHz). Hence, explain the statement that at very high frequency, an inductor in a circuit nearly amounts to an open circuit. How does an inductor behave in a dc circuit after the steady state?
Answer:
Inductance of the inductor, L = 0.5 Hz
Resistance of the resistor, R = 100 Ω
Potential of the supply voltage, V = 240 V
Frequency of the supply, v = 10 kHz = 10⁴ Hz
Angular frequency, ω = 2πv = 2 π × 10⁴ rad/s

(a) Peak voltage, V₀ = √2 × V = 240√2 V
Maximum current, I₀ = \(\frac{V_{0}}{\sqrt{R^{2}+\omega^{2} L^{2}}}\)
= \(\frac{240 \sqrt{2}}{\sqrt{(100)^{2}+\left(2 \pi \times 10^{4}\right)^{2} \times(0.50)^{2}}}\)
= 1.1 × 10^-2 A

(b) For phase difference, Φ, we have the relation
tanΦ = \(\frac{\omega L}{R}\) = \(\frac{2 \pi \times 10^{4} \times 0.5}{100}\) = 100π
Φ = 89.82° = \(\frac{89.82 \pi}{180}\) rad
ωt = \(\frac{89.82 \pi}{180}\)
t = \(\frac{89.82 \pi}{180 \times 2 \pi \times 10^{4}}\) = 25 μs

It can be observed that I₀ is very small in this case. Hence, at high frequencies, the inductor amounts to an open circuit.
In a dc circuit, after a steady state is achieved, ω = 0. Hence, inductor L behaves like a pure conducting object.

Question 15.
A 100 μF capacitor in series with a 40 Ω resistance is connected to a 110 V, 60 Hz supply.
(a) What is the maximum current in the circuit?
(b) What is the time lag between the current maximum and the voltage maximum?
Answer:
Capacitance of the capacitor, C = 100 μF = 100 × 10^-6 F = 10^-4 F
Resistance of the resistor, R = 40 Ω
Supply voltage, V = 110 V
Frequency of oscillations, v = 60 Hz
Angular frequency, co = 2πv = 2π × 60 rad/s = 120 π rad/s
For a RC circuit, we have the relation for impedance as
Z = \(\sqrt{R^{2}+\frac{1}{\omega^{2} C^{2}}}\)
peak voltage V₀ = V√2 = 110√2

(b) In an RC circuit, the voltage lags behind the current by a phase angle of Φ. This angle is given by the relation

= 1.55 × 10^-3 s
= 1.55 ms
Hence, the time lag between maximum current and maximum voltage is 1.55 ms.

Question 16.
Obtain the answers to (a) and (b) in Exercise 7.15 if the circuit is connected to a 110 V, 12 kHz supply? Hence, explain the statement that a capacitor is a conductor at very high frequencies. Compare this behaviour with that of a capacitor in a dc circuit after the steady state.
Answer:
Capacitance of the capacitor, C = 100 μF = 100 × 10^-6 F
Resistance of the resistor, R = 40 Ω
Supply voltage, V = 110 V
Frequency of the supply, v = 12 kHz = 12 × 10³ Hz
Angular frequency, ω = 2πv = 2 × π × 12 × 10³
= 24 π × 10³ rad/s
Peak voltage, V₀ = V√2 = 110 √2V

= 0.04 μs
Hence, Φ tends to become zero at high frequencies. At a high frequency, capacitor C acts as a conductor.
In a dc circuit, after the steady state is achieved, ω = 0. Hence, capacitor C acts an open circuit.

Question 17.
Keeping the source frequency equal to the resonating frequency of the series LCR circuit, if the three elements, L, C and R are arranged in parallel, show that the total current in the parallel LCR circuit is minimum at this frequency. Obtain the current rms value in each branch of the circuit for the elements and source specified in Exercise 7.11 for this frequency.
Answer:
Here, L = 5.0 H
C = 80 μF = 80 × 10^-6 F
R = 40Ω
The effective impedance of the parallel LCR is given by

Question 18.
A circuit containing a 80 mH inductor and a 60 µF capacitor in series is connected to a 230 V, 50 Hz supply. The resistance of the circuit is negligible.
(a) Obtain the current amplitude and rms values.
(b) Obtain the rms values of potential drops across each element.
(c) What is the average power transferred to the inductor?
(d) What is the average power transferred to the capacitor?
(e) What is the total average power absorbed by the circuit?
[‘Average’ implies ‘averaged over one cycle’.]
Answer:
Given,
V = 230 V, v = 50 Hz, L = 80 mH = 80 × 10^-3 H,
C = 60µF = 60 × 10^-6 F

(a) Inductive reactance X_L = ωL = 2πvL
= 2 × 3.14 × 50 × 80 × 10^-3
= 25.1 Ω

(b) RMS value of potential drops across L and C are
V_L = X_L I_rms = 25.1 × 8.23 = 207 V
V_C = X_C I_rms = 53.1 × 8.23 = 437 V
Net voltage = V_C – V_L = 230 V

(c) The voltage across L leads the current by angle \(\frac{\pi}{2}\) , therefore, average
power
P_av V_rms I_rms cos \(\frac{\pi}{2}\) = 0 (zero)

(d) The voltage across C lags behind the current by angle \(\frac{\pi}{2}\),
∴ p_av = V_rms I_rms cos \(\frac{\pi}{2}\) = 0

(e) As circuit contains pure I and pure C, average power consumed by LC circuit is zero.

Question 19.
Suppose the circuit in Exercise 7.18 has a resistance of 15 Ω. Obtain the average power transferred to each element of the circuit, and the total power absorbed.
Answer:
Here, R – 15Ω, L = 80 mH = 80 × 10^-3 H
C = 60 μF = 60 × 10^-6 F.
E_r.m.s. = 230 V
v = 50 Hz
> ω = 2πv = 2π × 50 =100 π
Z = impedance of LCR circuit
= \(\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}}\)

= 7.258 = 7.26 A
∴ Average power consumed by R or transferred to R is given by
(P_av)R = I²_r.m.s..R = (7.26)² × 15 = 790.614 W
= 791 W.
Also (P_av)_L and (P_av)_C be the average power transferred to I and C respectively.
(P_av)_L = E_r.m.s. . I_r.m.s. cosΦ
Here e.m.f. leads current by \(\frac{\pi}{2}\)
∴ (P_av)_L= E_r.m.s. . I_r.m.s. cos \(\frac{\pi}{2}\)
= 0
and (P_av )_C = = E_r.m.s. . I_r.m.s. cosΦ
= 0
( ∵ Φ = \(\frac{\pi}{2}\) and cos \(\frac{\pi}{2}\) = 0

If Pav be the total power absorbed in the circuit, then
P_av = (P_av)_L + (P_av )_C + (P_av )_R
= 0 + 0 + 791
= 791 W

Question 20.
A series LCR circuit with L = 0.12 H, C = 480 nF, R = 23 Ω is connected to a 230 V variable frequency supply.
(a) What is the source frequency for which current amplitude is maximum? Obtain this maximum value.
(b) What is the source frequency for which average power absorbed by the circuit is maximum? Obtain the value of this maximum power.
(c) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies?
(d) What is the Q-factor of the given circuit?
Answer:
Inductance, L = 0.12 H
Capacitance, C = 480 nF = 480 × 10^-9 F
Resistance, R = 23 Ω
Supply voltage, V = 230 V
Peak voltage is given as V₀ = √2V
V₀ = √2 × 230 = 325.22 V

(a) Current flowing in the circuit is given by the relation,
I₀ = \(\frac{V_{0}}{\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}}}\)
where, I₀ = maximum at resonance
At resonance, we have
ω_RL – \(\frac{1}{\omega_{R} C}[latex] = 0
where, ω_R = Resonance angular frequency
∴ ω_R = [latex]\frac{1}{\sqrt{L C}}\)
= \(\frac{1}{\sqrt{0.12 \times 480 \times 10^{-9}}}\)
= \(\frac{10^{5}}{\sqrt{12 \times 48}}=\frac{10^{5}}{24}\)
= 4166.67 rad/s
∴ Resonant frequency; v_R = \(\frac{\omega_{R}}{2 \pi}\) = \(\frac{4166.67}{2 \times 3.14}\) = 663.48 HZ
and, maximum current (I₀)_max = \(\frac{V_{0}}{R}\) = \(\frac{325.22}{23}\) 14.14 A

(b) Average power absorbed by the circuit is given as
P_av = \(\frac{1}{2}\)I₀²R

The average power is maximum at ω = ω₀ at which I₀ = (I₀)_max
∴ (p_av )_max = \(\frac{1}{2}\)(I₀)²_maxR
= \(\frac{1}{2}\) × (14.14)² × 23 = 2299.3 W
= 2300 W

(c) The power transferred to the circuit is half the power at resonant frequency.
Frequencies at which power transferred is half, ω = ω_R ± Δ ω
= 2π (v_R ± Δv)
where, Δω = \(\frac{R}{2 L}\)
= \(\frac{23}{2 \times 0.12}\) = 95.83 rad/s
Hence, change in frequency, Δ v = \(\frac{1}{2 \pi}\) Δω = \(\frac{95.83}{2 \pi}\) = 15.26 Hz
Thus power absorbed is half the peak power at
v_R + Δv = 663.48 + 15.26 = 678.74 Hz
and, v_R ΔV = 663.48 – 15.26 = 648.22 Hz
Hence, at 648.22 Hz and 678.74 Hz frequencies, the power transferred is half.
At these frequencies, current amplitude can be given as
I’ = \(\frac{1}{\sqrt{2}}\) × (I₀)max = \(\frac{14.14}{\sqrt{2}}=\frac{14.14}{1.414}\) = 10 A

(d) Q-factor of the given circuit can be obtained using the relation,
Q = \(\frac{\omega_{R} L}{R}\) = \(\frac{4166.67 \times 0.12}{23}\) = 21.74
Hence, the Q-factor of the given circuit is 21.74.

Question 21.
Obtain the resonant frequency and Q-factor of a series LCR circuit with L = 3.0 H, C = 27 μF and R = 7.4 Ω. It is desired to improve the sharpness of the resonance of the circuit by reducing its ‘full width at half maximum’ by a factor of 2. Suggest a suitable way.
Answer:
Inductance, L = 3.0 H
Capacitance, C = 27 μF = 27 × 10^-6F
Resistance, R = 7.4 Ω
At resonance, resonant frequency of the source for the given LCR series circuit is given as
ω_r = \(\frac{1}{\sqrt{L C}}\) = \(\frac{1}{\sqrt{3 \times 27 \times 10^{-6}}}\)
\(\frac{10^{3}}{9}\) = 111.11 rad s^-1
Q-factor of the series
Q = \(\frac{\omega_{r} L}{R}\) = \(\frac{111.11 \times 3}{7.4}\) = 45.0446
To improve the sharpness of the resonance by reducing its ‘full width at half maximum’ by a factor of 2 without changing cor, we need to reduce R to half i. e., Resistance = \(\frac{R}{2}=\frac{7.4}{2}\) = 3.7 Ω.

Question 22.
Answer the following questions :
(a) In any ac circuit, is the applied instantaneous voltage equal to the algebraic sum of the instantaneous voltages across the series elements of the circuit? Is the same true for rms voltage?

(b) A capacitor is used in the primary circuit of an induction coil.

(c) An applied voltage signal consists of a superposition of a dc voltage and an ac voltage of high frequency. The circuit consists of an inductor and a capacitor in series. Show that the dc signal will appear across C and the ac signal across L.

(d) A choke coil in series with a lamp is connected to a dc line. The lamp is seen to shine brightly. Insertion of an iron core in the choke causes no change in the lamp’s brightness. Predict the corresponding observations if the connection is to an ac line.

(e) Why is choke coil needed in the use of fluorescent tubes with ac mains? Why can we not use an ordinary resistor instead of the choke coil?
Answer:
(a) Yes; the statement is not true for rms voltage.
It is true that in any ac circuit, the applied voltage is equal to the average sum of the instantaneous voltages across the series elements of the circuit. However, this is not true for rms voltage because voltages across different elements may not be in phase.

(b) High induced voltage is used to charge the capacitor.
A capacitor is used in the primary circuit of an induction coil. This is because when the circuit is broken, a high induced voltage is used to charge the capacitor to avoid sparks.

(c) The dc signal will appear across capacitor C because for dc signals, the impedance of an inductor (L) is negligible while the impedance of a capacitor (C) is very high (almost infinite). Hence, a dc signal appears across C. For an ac signal of high frequency, the impedance of L is high and that of C is very low. Hence, an ac signal of high frequency appears across L.

(d) If an iron core is inserted in the choke coil (which is in series with a lamp connected to the ac line), then the lamp will glow dimly. This is because the choke coil and the iron core increase the impedance of the circuit.

(e) A choke coil is needed in the use of fluorescent tubes with ac mains because it reduces the voltage across the tube without wasting much power. An ordinary resistor cannot be used instead of a choke coil for this purpose because it wastes power in the form of heat.

Question 23.
A power transmission line feeds input power at 2300 V to a stepdown transformer with its primary windings having 4000 turns. What should be the number of turns in the secondary in order to get output power at 230 V?
Answer:
Input voltage, V₁ = 2300 V
Number of turns in primary coil, n₁ = 4000
Output voltage, V₂ = 230 V
Number of turns in secondary coil = n₂
Voltage is related to the number of turns as
\(\frac{V_{1}}{V_{2}}=\frac{n_{1}}{n_{2}}\)
\(\frac{2300}{230}=\frac{4000}{n_{2}}\)
n₂ = \(\frac{4000 \times 230}{2300}\) = 400
Hence, there are 400 turns in the second winding.

Question 24.
At a hydroelectric power plant, the water pressure head is at a height of 300 m and the water flow available is 100 m³s^-1 . If the turbine generator efficiency is 60%, estimate the electric power available from the plant (g = 9.8 ms^-2).
Answer:
Height of the water pressure head, h = 300 m
Volume of water flow per second, V = 100 m³/s
Efficiency of turbine generator, η = 60% = 0.6
Acceleration due to gravity, g = 9.8 m/ s²
Density of water, ρ = 10³ kg/m³
Electric power available from the plant = η × h ρ gV
= 0.6 × 300 × 10³ × 9.8 × 100
= 176.4 × 10⁶ W
= 176.4 MW

Question 25.
A small town with a demand of 800 kW of electric power at 220 V is situated 15 km away from an electric plant generating power at 440 V. The resistance of the two wire line carrying power is 0.5 Ω per km. The town gets power from the line through a 4000-220 V step-down transformer at a sub-station in the town.
(a) Estimate the line power loss in the form of heat.
(b) How much power must the plant supply, assuming there is negligible power loss due to leakage?
(c) Characterise the step up transformer at the plant.
Answer:
Total electric power required, P = 800 kW = 800 × 10³ W
Supply voltage, V = 220 V
Voltage at which electric plant is generating power, V’ = 440 V
Distance between the town and power generating station, d = 15 km
Resistance of the two wire lines carrying power = 0.5 Ω/km
Total resistance of the wires, R = (15 + 15)0.5 = 15Ω
A step-down transformer of rating 4000 – 220 V is used in the sub-station.
Input voltage, V₁ = 4000 V
Output voltage, V₂ = 220 V
rms current in the wire lines is given as
I = \(\frac{P}{V_{1}}\) = \(\frac{800 \times 10^{3}}{4000}\) = 200 A

(a) Line power loss = I²R = (200)² × 15 = 600 × 10³ W = 600 kW

(b) Assuming that the power loss is negligible due to the leakage of the current.
Total power supplied by the plant = 800 kW + 600 kW = 1400 kW

(c) Voltage drop in the power line = IR = 200 × 15 = 3000 V
Hence, total voltage transmitted from the plant = 3000 + 4000 = 7000 V Also, the power generated is 440 V.
Hence, the rating of the step-up transformer situated at the power plant is 440 V – 7000 V.

Question 26.
Do the same exercise as above with the replacement of the earlier transformer by a 40,000-220 V step-down transformer (Neglect, as before, leakage losses though this may not be a good assumption any longer because of the very high voltage transmission involved). Hence, explain why high voltage transmission is preffered?
Answer:
The rating of the step-down transformer is 40000 V – 220 V
Input voltage, V₁ = 40000 V
Output voltage, V₂ = 220 V
Total electric power required, P = 800 kW = 800 × 10³ W
Source potential, V = 220 V
Voltage at which the electric plant generates power, V’ = 440 V
Distance between the town and power generating station, d = 15 km
Resistance of the two wire lines carrying power = 0.5 Ω/km
Total resistance of the wire lines, R = (15 + 15)0.5 = 15 Ω
rms current in the wire line is given as
I = \(\frac{P}{V_{1}}\) = \(\frac{800 \times 10^{3}}{40000}\) = 20A

(a) Line power loss = I²R
= (20)² × 15 = 6000 W = 6 kW

(b) Assuming that the power loss is negligible due to the leakage of current.
Hence, total power supplied by the plant = 800 kW + 6 kW = 806 kW

(c) Voltage drop in the power line = 7R = 20 × 15 = 300 V
Hence, voltage that is transmitted by the power plant
= 300 + 40000 = 40300 V
The power is being generated in the plant at 440 V.
Hence, the rating of the step-up transformer needed at the plant is 440 V – 40300 V. ‘
Hence, power loss during transmission = \(\frac{600}{1400}\) x 100 = 42.8%
In the previous exercise, the power loss due to the same reason is
\(\frac{6}{800}\) × 100 = 0.744%
Since the power loss is less for a high voltage transmission, high voltage transmissions are preferred for this purpose.

Punjab State Board PSEB 12th Class Physics Book Solutions Chapter 9 Ray Optics and Optical Instruments Textbook Exercise Questions and Answers.

PSEB Solutions for Class 12 Physics Chapter 9 Ray Optics and Optical Instruments

PSEB 12th Class Physics Guide Ray Optics and Optical Instruments Textbook Questions and Answers

Question 1.
A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius of curvature 36 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image? Describe the nature and size of the image. If the candle is moved closer to the mirror, how would the screen have to be moved?
Answer:
Size of the candle, h = 2.5 cm
Image size = h’
Object distance, u = -27 cm
Radius of curvature of the concave mirror, R = -36 cm
Focal length of the concave mirror, f = \(\frac{R}{2}=\frac{-36}{2}\) = -18 cm
Image distance = v

The image distance can be obtained using the mirror formula

The negative sign shows that the image is formed in front of the mirror i.e., on the side of the object itself. Thus the screen must be placed at a distance of 54 cm in front of the mirror.
The magnification of the image is given as

The height of the candle’s image is 5 cm. The negative sign indicates that the image is inverted and virtual.
If the candle is moved closer to the mirror, then the screen will have to be moved away from the mirror in order to obtain the image.

Question 2.
A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror.
Answer:
Given u = -12 cm, f = +15 cm. (convex mirror)

That is image is formed at a distance of 6.67 cm behind the mirror.
Magnification m = \(-\frac{v}{u}=-\frac{\frac{20}{3}}{-12} \) = \(\frac{5}{9}\)
Size of image I = mO = \(\frac{5}{9}\) x 4.5 = 2.5 cm
The image is erect, virtual and has a size 2.5 cm.

Its position is 6.67 cm behind the mirror when needle is moved farther, the image moves towards the focus and its size goes on decreasing.

Question 3.
A tank is filled with water to a height of 12.5 cm. The apparent depth of a needle lying at the bottom of the tank is measured by a microscope to be 9.4 cm. What is the refractive index of water? If water is replaced by a liquid of refractive index 1.63 up to the same height, by what distance would the microscope have to be moved to focus on the needle again?
Answer:
Case I:
When tank is filled with water Actual depth of the needle in water, h₁ = 12.5cm
Apparent depth of the needle in water, h₂ =9.4cm
Refractive index of water = μ
The value μ can be obtained as follows
μ = \(\frac{\text { Actual depth }}{\text { Apparent depth }}\)
= \(\frac{h_{1}}{h_{2}}=\frac{12.5}{9.4}\) ≈ 1.33
Hence, the refractive index of water is about 1.33

Case II: When tank is filled with liquid
Water is replaced by a liquid of refractive index, μ’ = 1.63
The actual depth of the needle remains the same, but its apparent depth changes.
Let y be the new apparent depth of the needle. Hence, we can write the relation
μ’ = \(\frac{h_{1}}{y}\)
y = \(\frac{h_{1}}{\mu^{\prime}}=\frac{12.5}{1.63}\) = 7.67 cm
Hence, the new apparent depth of the needle is 7.67cm. It is less than h2 Therefore, to focus the needle again, the microscope should be moved up. Distance by which the microscope should be moved up =9.4-7.67 = 1.73 cm.

Question 4.
Figures 9.34 (a) and (b) show refraction of a ray in air incident at 60° with the normal to a glass air and water-air interface, respectively. Predict the angle of refraction in glass when the angle of incidence in water is 45° with the normal to a water-glass interface [Fig. 9.34 (c)]

Answer:
As per the given figure, for the glass-air interface
Angle of incidence, i = 60°
Angle of refraction, r = 35°
The relative refractive index of glass with respect to air is given by Snell’s law as
^aμ_g = \( \frac{\sin i}{\sin r}\)
= \(\frac{\sin 60^{\circ}}{\sin 35^{\circ}}=\frac{0.8660}{0.5736}\) = 1.51 …………………….. (1)
As per the given figure, for the air-water interface
Angle of incidence, j = 600
Angle of refraction, r = 470
The relative refractive index of water with respect to air is given by Snell’s law as
^wμ_w = \( \frac{\sin i}{\sin r}\)
= \(\frac{\sin 60^{\circ}}{\sin 47^{\circ}}=\frac{0.8660}{0.7314}\) = 1.184 …………………………… (2)

Using equations (1) and (2), the relative refractive index of glass with respect to water can be obtained as
^wμ_g = \(\frac{a_{g}}{a_{w_{w}}}\)
= \( \frac{1.51}{1.184} \) = 1.275

The following figure shows the situation involving the glass-water interface

Angle of incidence, i = 45
Angle of reflection = r
From Snell’s law, r can be calculated as, \(\frac{\sin i}{\sin r}\) = ^wμ_g
\(\frac{\sin 45^{\circ}}{\sin r}\) = 1.275
sin r = \(\frac{\frac{1}{\sqrt{2}}}{1.275}=\frac{0.707}{1.275}\) = 0.5546
r = sin^-1(0.5546) = 38.68°
Hence, the angle of refraction at the water-glass interface is 38.68°

Question 5.
A small bulb is placed at the bottom of a tank containirg water to a depth of 80 cm. What is the area of the surface of water through which light from the bulb can emerge out? Refractive index of water is 1.33 (Consider the bulb to be a point source.)
Answer:
Actual depth of the bulb in water, d₁ = 80 cm = 0.8 m
Refractive index of water, μ = 1.33
The given situation is shown in the following figure

where,
i = Angle of Incidence
r = Angle of Refraction = 90°
Since the bulb is a point source, the emergent light can be considered as a circle of radius,
R = \(\frac{A C}{2}\) = AO = OC
Using Snell’s law, we can write the relation for the refractive index of water as
μ = \(\frac{\sin r}{\sin i}\)
1.33 = \(\frac{\sin 90^{\circ}}{\sin i}\)
i = sin^-1\(\left(\frac{1}{1.33}\right)\) = 48.75°

Using the given figure, we have the relation
tan i = \(\frac{O C}{O B}=\frac{R}{d_{1}}\)
∴R = tan 48.75° x 0.8 = 0.91 m
∴ Area of the surface of water = πR²
= π(0.91)²
= 2.61 m²
Hence, the area of the surface of water through which the light from the bulb can emerge is approximately 2.61 m².

Question 6.
A prism is made of glass of unknown refractive index. A parallel beam of light is incident on a face of the prism. The angle of minimum deviation is measured to be 40°. What is the refractive index of the material of the prism? The refracting angle of the prism is 60°. If the prism is placed in water (refractive index 1.33), predict the new angle of minimum deviation of a parallel beam of light.
Answer:
Angle of minimum deviation, δ_m = 40 °
Refracting angle of the prism, A = 60°
Refractive index of water, μ = 1.33
Let μ’ be the refractive index of the material of the prism.
The angle of deviation and refracting angle of the prism are related to refractive index (μ’) as
μ’ = \(\frac{\sin \left(\frac{A+\delta_{m}}{2}\right)}{\sin \left(\frac{A}{2}\right)} \)
= \(\frac{\sin \left(\frac{60^{\circ}+40^{\circ}}{2}\right)}{\sin \left(\frac{60^{\circ}}{2}\right)}=\frac{\sin 50^{\circ}}{\sin 30^{\circ}}=\frac{0.766}{0.5}\)
= 1.532
Hence, the refractive index of the material of the prism is 1.532.
Since the prism is placed in water, let 8 ^ be the new angle of minimum deviation for the same prism.

The refractive index of glass with respect to water is given by the relation


Hence, the new minimum angle of deviation is 10.32°.

Question 7.
Double-convex lenses are to be manufactured from a glass of refractive index 1.55, with both faces of the same radius of curvature. What is the radius of curvature required if the focal length is to be 20 cm?
Answer:
Lens maker formula is
\(\frac{1}{f}=(n-1)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)\) …………………………………… (1)
If R is radius of curvature of double convex lens, then,

∴ R = 2(n-1)f
Here, n =1.55, f = +20 cm
∴ R = 2 (1.55 -1) x 20 = 22 cm

Question 8.
A beam of light converges at a point P. Now a lens is placed in the path of the convergent beam 12 cm from P. At what point does the beam converge if the lens is (a) a convex lens of focal length 20 cm, and (b) a concave lens of focal length 16 cm?
Answer:
In the given situation, the object is virtual and the image formed is real.
Object distance, u = +12cm
(a) Focal length of the convex lens, f = 20 cm
Image distance = v
According to the lens formula, we have the relation

∴ v = \(\frac{60}{8}\) = 7.5cm
Hence, the image is formed 7.5cm away from the lens, toward its right.

(b) Focal length of the concave lens, f = -16 cm
Image distance = v
According to the lens formula, we have the relation

∴ v = 48 cm
Hence, the image is formed 48 cm away from the lens, toward its right.

Question 9.
An object of size 3.0 cm is placed 14 cm in front of a concave lens of focal length 21 cm. Describe the image produced by the lens. What happens if the object is moved further away from the lens?
Answer:
Size of object O = 3.0 cm
u = -14 cm, f = -21 cm (concave lens)
∴ Formula \(\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\)
⇒ \(\frac{1}{v}=\frac{1}{f}+\frac{1}{u}\)
or \(\frac{1}{v}=\frac{1}{-21}+\frac{1}{-14}=-\frac{2+3}{42}\)
or v = \(-\frac{42}{5}\) = -8.4 cm
Size of image I = \(\frac{v}{u}\) O
= \(\frac{-8.4}{-14}\) x 3.0 cm = 1.8 cm

That is, image is formed at a distance of 8.4 cm in front of lens. The image is virtual, erect and of size 1.8 cm. As the object is moved farther from the lens, the image goes on shifting towards focus and its size goes on decreasing. The image is never formed beyond the focus of the concave lens.

Question 10.
What is the focal length of a convex lens of focal length 30 cm in contact with a concave lens of focal length 20 cm? Is the system
a converging or a diverging lens? Ignore thickness of the lenses.
Answer:
Given f₁ = +30 cm, f₂ = -20 cm
The focal length (F) of combination is given by
\(\frac{1}{F}=\frac{1}{f_{1}}+\frac{1}{f_{2}}\)
⇒ F = \(\frac{f_{1} f_{2}}{f_{1}+f_{2}}\)
= \(\frac{30 \times(-20)}{30-20}\) = -60 cm
That is, the focal length of combination is 60 cm and it acts like a diverging lens.

Question 11.
A compound microscope consists of an objective lens of focal length 2.0 cm and an eyepiece of focal length 6.25 cm separated by a distance of 15 cm. How far from the objective should an object be placed in order to obtain the final image at (a) the least distance of distinct vision (25cm), and (b) at infinity? What is the magnifying power of the microscope in each case?
Answer:
Focal length of the objective lens, f₀ = 2.0 cm
Focal length of the eyepiece, f_e = 6.25cm
Distance between the objective lens and the eyepiece, d = 15cm
(a) Least distance of distinct vision, d’ = 25cm
∴ Image distance for the eyepiece, v_e = -25cm
Object distance for the eyepiece = u_e
According to the lens formula, we have the relation
\(\frac{1}{v_{e}}-\frac{1}{u_{e}}=\frac{1}{f_{e}}\)
or \(\frac{1}{u_{e}}=\frac{1}{v_{e}}-\frac{1}{f_{e}}\)
= \(\frac{1}{-25}-\frac{1}{6.25}=\frac{-1-4}{25}=\frac{-5}{25}\)
∴ u_e = -5cm
Image distance for the objective lens, v₀ = d + ue =15-5 = 10 cm
Object distance for the objective lens = u₀
According to the lens formula, we have the relation
\(\frac{1}{v_{o}}-\frac{1}{u_{o}}=\frac{1}{f_{o}}\)
\(\frac{1}{u_{0}}=\frac{1}{v_{0}}-\frac{1}{f_{0}}=\frac{1}{10}-\frac{1}{2}=\frac{1-5}{10}=\frac{-4}{10}\)
∴ u₀=-2.5cm
Magnitude of the object distance, |u₀| = 2.5 cm
The magnifying power of a compound microscope is given by the relation
m = \(\frac{v_{o}}{\left|u_{o}\right|}\left(1+\frac{d^{\prime}}{f_{e}}\right)\)
= \(\frac{10}{2.5}\left(1+\frac{25}{6.25}\right)\) = 4(1+4) = 20
Hence, the magnifying power of the microscope is 20.

(b) The final image is formed at infinity.
∴ Image distance for the eyepiece, v_e = ∞
Object distance for the eyepiece = u_e
According to the lens formula, we have the relation
\(\frac{1}{v_{o}}-\frac{1}{u_{o}}=\frac{1}{f_{o}}\)
\(\frac{1}{u_{o}}=\frac{1}{v_{o}}-\frac{1}{f_{o}}=\frac{1}{8.75}-\frac{1}{2.0}=\frac{2-8.75}{17.5}\)
∴ u₀ = \(\frac{17.5}{6.75}\) = -2.59 cm
Magnitude of the object distance, |u₀| = 2.59 cm
The magnifying power of a compound microscope is given by the relation
m = \(\frac{v_{o}}{\left|u_{o}\right|}\left(1+\frac{d^{\prime}}{f_{e}}\right)\)
= \(\frac{8.75}{2.59} \times\left(1+\frac{25}{6.25}\right)\) = 13.51
Hence, the magnifying power of the microscope is 13.51.

Question 12.
A person with a normal near point (25cm) using a compound microscope with objective of focal length 8.0 mm and an eyepiece of focal length 2.5 cm can bring an object placed at 9.0 mm from the objective in sharp focus. What is the separation between the two lenses? Calculate the magnifying power of the microscope.
Answer:
Focal length of the objective lens, f₀= 8 mm = 0.8cm
Focal length of the eyepiece, f_e = 2.5 cm
Object distance for the objective lens, u₀ = -9.0 mm = -0.9 cm
Least distance of distant vision, d = 25 cm
Image distance for the eyepiece, v_e = -d = -25 cm
Object distance for the eyepiece = u_e

Using the lens formula, we can obtain the value of u_e as

∴ u_e = \(-\frac{25}{11}\) = -2.27 cm
We can also obtain the value of the image distance for the objective lens (v₀) using the lens formula.

∴ v₀ = 7.2 cm
The distance between the objective lens and the eyepiece = |u_e|+v₀
= 2.27+ 7.2 = 9.47cm
The magnifying power of the microscope is calculated as \(\frac{v_{o}}{\left|u_{o}\right|}\left(1+\frac{d}{f_{e}}\right)\)
= \(\frac{7.2}{0.9}\left(1+\frac{25}{2.5}\right)\)
= 8(1 +10) = 88
Hence, the magnifying power of the microscope is 88.

Question 13.
A small telescope has an objective lens of focal length 144 cm and an eyepiece of focal length 6.0cm. What is the magnifying power of the telescope? What is the separation between the objective and the eyepiece?
Answer:
Focal length of the objective lens, f₀ = 144 cm
Focal length of the eyepiece, f_e = 6.0 cm
The magnifying power of the telescope is given as, m = \(\frac{f_{o}}{f_{e}}=\frac{144}{6}\) = 24
The separation between the objective lens and the eyepiece is calculated as
= f_o + f_e
= 144 + 6 = 150 cm
Hence, the magnifying power of the telescope is 24 and the separation between the objective lens and the eyepiece is 150 cm.

Question 14.
(a) A giant refracting telescope at an observatory has an objective lens of focal length 15 m. If an eyepiece of focal length 1.0 cm is used, what is the angular magnification of the telescope?
(b) If this telescope is used to view the moon, what is the diameter of the image of the moon formed by the objective lens? The diameter of the moon is 3.48 x 10⁶ m, and the radius of lunar orbit is 3.8 x 10⁸ m.
Answer:
(a) Given f₀ = 15 m,
f_e = 1.0 cm = 1.0 x 10^-2 m
Angular magnification of telescope,
m = \(-\frac{f_{o}}{f_{e}}=-\frac{15}{1.0 \times 10^{-2}}\) = -1500
Negative sign shows that the final image is inverted.
(b) Let D be diameter of moon, d diameter of image of moon formed by objective and r be the distance of moon from objective lens, then

Question 15.
Use the mirror equation to deduce that:
(a) an object placed between f and 2f of a concave mirror produces a real image beyond 2f.
(b) a convex mirror always produces a virtual image independent of the location of the object.
(c) the virtual image produced by a convex mirror is always diminished In size and is located between the focus and the
pole.
(d) an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image.
[Note: This exercise helps you deduce algebraically properties of images that one obtains from explicit ray diagrams.]
Answer:
(a) For a concave mirror, the focal length (f) is negative
∴ f<o
When the object is placed on the left side of the mirror, the object distance (u) is negative.
∴ u<O
For image distance v, we can write the mirror formula
\(\frac{1}{v}=\frac{1}{f}-\frac{1}{u}\) …………………………………… (1)
The object lies between f and 2f.
∴ 2f < u < f (∵ u and f are negative) ∴ \(\frac{1}{2 f}>\frac{1}{u}>\frac{1}{f}\)
\(-\frac{1}{2 f}<-\frac{1}{u}<-\frac{1}{f}\)
\(\frac{1}{f}-\frac{1}{2 f}<\frac{1}{f}-\frac{1}{u}<0\) ………………………………… (2)
Using equation (1), we get
\(\frac{1}{2 f}<\frac{1}{v}<0\)

∴ \(\frac{1}{v}\) is negative, i.e., v is negative.
\(\frac{1}{2 f}<\frac{1}{v}\) 2f > v
-v > -2 f
Therefore, the image lies beyond 2f.

(b) For a convex mirror, the focal length (f) is positive.
∴ f>o
When the object is placed on the left side of the mirror, the object distance (u) is negative.
∴ u<O
For image distance y, we have the mirror formula
\(\frac{1}{v}=\frac{1}{f}-\frac{1}{u}\)
Using equation (2), we can conclude that
\(\frac{1}{\nu}\) < 0 v v> 0
Thus, the image is formed on the back side of the mirror.
Hence, a convex mirror always produces a virtual image, regardless of the object distance.

(c) For a convex mirror, the focal length (f) is positive.
∴ f> 0
When the object is placed on the left side of the mirror, the object distance (u) is negative.
∴ u< 0
For image distance v, we have the mirror formula
\(\frac{1}{v}=\frac{1}{f}-\frac{1}{u}\)
But we have u < 0 ∴ \(\frac{1}{v}>\frac{1}{f}\)
v < f
Hence, the image formed is diminished and is located between the focus (f) and the pole.

(d) For a concave mirror, the focal length (f) is negative.
∴ f< 0
When the object is placed on the left side of the mirror, the object distance (u) is negative.
∴ u< 0 It is placed between the focus (f) and the pole. ∴f > u > 0
\(\frac{1}{f}<\frac{1}{u}\) < 0 \(\frac{1}{f}-\frac{1}{u}\) > 0
For image distance v, we have the mirror formula

The image is formed on the right side of the mirror. Hence, it is a virtual image.
For u < 0 and v > 0, we can write
\(\frac{1}{u}>\frac{1}{v}\)
v > u
Magnification, m = \(\frac{v}{u}\) > 1 u
Hence, the formed image is enlarged.

Question 16.
A small pin fixed on a tabletop is viewed from above from a distance of 50 cm. By what distance would the pin appear to be raised if it is viewed from the same point through a 15 cm thick glass slab held parallel to the table? Refractive index of glass = 1.5. Does the answer depend on the location of the slab?
Answer:
Actual depth of the pin, d = 15cm
Apparent depth of the pin = d’
Refractive index of glass, µ = 1.5

Ratio of actual depth to the apparent depth is equal to the refractive index of glass, i.e.
µ = \(\frac{d}{d^{\prime}}\)
∴ d’ = \(\frac{d}{\mu}\)
= \(\frac{15}{1.5}\) = 10 cm
The distance at which the pin appears to be raised = d-d’=15-10 = 5 cm
For a small angle of incidence, this distance does not depend upon the location of the slab.

Question 17.
(a) Figure 9.35 shows a cross-section of a ‘light pipe’ made of a glass fibre of refractive index 1.68. The outer covering of the pipe is made of a material of refractive index 1.44. What is the range of the angles of the incident rays with the axis of the pipe for which total reflections inside the pipe take place, as shown in the figure.
(b) What is the answer if there is no outer covering of the pipe?

Answer:
(a) Refractive index of the glass fibre, µ₂ = 1.68
Refractive index of the outer covering of the pipe, µ₁ =1.44
Angle of incidence = i
Angle of refraction = r
Angle of incidence at the interface = i’

The refractive index (µ) of the inner core-outer core interface is given as
µ = \(\frac{\mu_{2}}{\mu_{1}}=\frac{1}{\sin i^{\prime}}\)
sin i’ = \(\frac{\mu_{1}}{\mu_{2}}=\frac{1.44}{1.68}\) = 0.8571
∴ i’ = 59°

For the critical angle, total internal reflection (TIR) takes place only when i > i’. i.e., i > 59°
Maximum angle of reflection, r_max = 90°-i’ = 90°-59°= 31°
Let, i_max be the maximum angle of incidence.
The refractive index at the air – glass interface, µ₂ =1.68
µ₂ = \(\frac{\sin i_{\max }}{\sin r_{\max }}\)
sin i_max = µ₂ sin r_max = 1.68 sin31°
= 1.68 x 0.5150
= 0.8652
∴i_max = sin^-1 (0.8652) ≈ 60°
Thus, all the rays incident at angles lying in the range 0 < i < 60° will suffer total internal reflection.

(b) If the outer covering of the pipe is not present, then
Refractive index of the outer pipe, µ₁ = Refractive index of air = 1
For the angle of incidence i =90°, we can write Snell’s law at the air-pipe interface as
\(\frac{\sin i}{\sin r}\) = µ₂ = 1.68
sin r = \(\frac{\sin 90^{\circ}}{1.68}=\frac{1}{1.68}\)
r = sin^-1(0.5952)
∴ i’ = 90°-36.5°= 53.5°
Since i’ > r, all incident rays will suffer total internal reflection.

Question 18.
Answer the following questions:
(a) You have learnt that plane and convex mirrors produce virtual images of objects. Can they produce real images under some circumstances? Explain.
(b) A virtual image, we always say, cannot be caught on a screen. Yet when we ‘see’ a virtual image, we are obviously bringing it onto the ‘screen’(i.e., the retina) of our eye. Is there a contradiction?
(c) A diver underwater, looks obliquely at a fisherman standing on the bank of a lake. Would the fisherman look taller or shorter to the diver than what he actually is?
(d) Does the apparent depth of a tank of water change if viewed obliquely? If so, does the apparent depth increase or decrease?
(e) The refractive index of diamond is much greater than that – of ordinary glass. Is this fact of some use to a diamond cutter?
Answer:
(a) Yes, they produce real images under some circumstances. If the object is virtual, i.e., if the light rays converging at a point behind a plane mirror (or a convex mirror) are reflected to a point on a screen placed in front of the mirror, then a real image will be formed.

(b) No, there is no contradiction. A virtual image is formed when light rays diverge. The convex lens of the eye causes these divergent rays to converge at the retina. In this case, the virtual image serves as an object for the lens to produce a real image.

(c) The diver is in the water and the fisherman is on land (i.e., in the air). Water is a denser medium than air. It is given that the diver is viewing the fisherman. This indicates that the light rays are traveling from a denser medium to a rarer medium. Hence, the refracted rays will move away from the normal. As a result, the fisherman will appear to be taller.

(d) Yes, the apparent depth of a tank of water changes when viewed obliquely. This is because light bends on traveling from one medium to another. The apparent depth of the tank, when viewed obliquely, is less than the near-normal viewing.

(e) Yes, the refractive index of diamond (2.42) is more than that of ordinary glass (1.5). The critical angle for diamond is less than that for glass. A diamond cutter uses a large angle of incidence to ensure that the light entering the diamond is totally reflected from its faces. This is the reason for the sparkling effect of a diamond.

Question 19.
The image of a small electric bulb on the wall of a room is to be obtained on the opposite wall 3 m away by means of a large convex lens. What is the maximum possible focal length of the lens required for the purpose? ’’
Answer:
Here, u + v = 3 m, :.v = 3 -u
From lens formula,

or u = \(\frac{3 \pm \sqrt{3^{2}-4.3 f}}{2}\)
For real solution, 9 -12, f should be positive.
It., 9 -12f > 0
or 9 >12f.
or f < \(\frac{9}{12}\) < \(\frac{3}{4}\) m
∴ The maximum focal length of the lens required for the purpose is \(\frac{3}{4}\) m
i.e, f_max = 0.7 m

Question 20.
A screen is placed 90 cm from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by 20 cm. Determine the focal length of the lens.
Answer:
Here, O is a position of object and I is position of image (screen).
Distance OI = 90 cm
L₁ and L₂ are the two positions of the lens.
∴ Distance between L₁ and L₂ = O₁ O₂ = 20 cm
For Position L₁ of the Lens: Let x be the distance of the object from the lens.
∴ u₁ = -x
∴ Distance of the image from the lens, v₁ = +(90 – x)

If f be the focal length of the lens, then using lens formula,
\(-\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\) we get
\(-\frac{1}{-x}+\frac{1}{90-x}=\frac{1}{f}\)
or \(\frac{1}{f}=\frac{1}{x}+\frac{1}{90-x}\) ……………………………….. (1)
For Position L₂ of the Lens : Let u₂ and v₂ be the distances of the object and image from the lens in this position.
∴ u₂=-(X + 20),
v₂ = +[90-(x+20)] = +(70-x)
∴ Using lens formula,

Question 21.
(a) Determine the ‘effective focal length’ of the combination of the two lenses in Exercise 9.10, if they are placed 8.0 cm apart with their principal axes coincident. Does the answer depend on which side of the combination a beam of parallel light is incident? Is the notion of effective focal length of this system useful at all?

(b) An object 1.5 cm in size is placed on the side of the convex lens in the arrangement (a) above. The distance between the object and the convex lens is 40 cm. Determine the magnification produced by the two-lens system and the size of the image.
Answer:
Focal length of the convex lens, f₁ = 30 cm
Focal length of the concave lens,f₂ = -20 cm
Distance between the two lenses, d = 8.0 cm

(a)
(i) When the parallel beam of light is incident on the convex lens first.
According to the lens formula, we have
\(\frac{1}{v_{1}}-\frac{1}{u_{1}}=\frac{1}{f_{1}}\)
where, μ₁ = Object distance = ∞, v₁ = Image distance = ?
\(\frac{1}{v_{1}}=\frac{1}{30}-\frac{1}{\infty}=\frac{1}{30}\)
∴ v₁ = 30 cm
The image will act as a virtual object for the concave lens. Applying lens formula to the concave lens, we have
\(\frac{1}{v_{2}}-\frac{1}{u_{2}}=\frac{1}{f_{2}}\)
where, u₂ = Object distance = (30 – d) = 30 – 8 = 22 cm,
v₂ = Image distance=?
\(\frac{1}{v_{2}}=\frac{1}{22}-\frac{1}{20}=\frac{10-11}{220}=\frac{-1}{220}\)
∴ v₂ = -220 cm
The parallel incident beam appears to diverge from a point that is \(\left(220-\frac{d}{2}=220-\frac{8}{2}=220-4=216 \mathrm{~cm}\right)\) from the centre of the combination of the two lenses.

(ii) When the parallel beam of light is incident, on the concave lens first. According to the lens formula, we have
\(\frac{1}{v_{2}}-\frac{1}{u_{2}}=\frac{1}{f_{2}}\)
\(\frac{1}{v_{2}}=\frac{1}{f_{2}}+\frac{1}{u_{2}}\)
where, u₂ = Object distance = -∞, v₂ = Image distance = ?
\(\frac{1}{v_{2}}=\frac{1}{-20}+\frac{1}{-\infty}=-\frac{1}{20}\)
∴ v₂ = -20 cm
The image will act as a real object for the .convex lens.
Applying lens formula to the convex lens, we have
\(\frac{1}{v_{1}}-\frac{1}{u_{1}}=\frac{1}{f_{1}}\)
where, u₁ = Object distance = -(20 + d) = -(20 + 8) = -28 cm v₁ = Image distance = ?
\(\frac{1}{v_{1}}=\frac{1}{30}+\frac{1}{-28}=\frac{14-15}{420}=\frac{-1}{420}\)
∴ v₁ = -420 cm
Hence, the parallel incident beam appear to diverge from a point that is (420 – 4 = 416 cm) from the left of the centre of the combination of the two lenses. The answer depends on the side of the combination at which the parallel beam of light is incident. The notion of effective focal length does not seem to be useful for this combination.

(b) Height of the object, h₁ =1.5 cm
Object distance from the side of the convex lens, u₁ = -40 cm
|u_i| = 40 cm

According to the lens formula
\(\frac{1}{v_{1}}-\frac{1}{u_{1}}=\frac{1}{f_{1}}\)
where, v₁ = Image distance =?
\(\frac{1}{v_{1}}=\frac{1}{30}+\frac{1}{-40}=\frac{4-3}{120}=\frac{1}{120}\)
∴ v₁ = 120 cm
Magnification, m= \(\frac{v_{1}}{\left|u_{1}\right|}=\frac{120}{40}\) = 3

Hence, the magnification due to the convex lens is 3.
The image formed by the convex lens acts as an object for the concave lens.
According to the lens formula
\(\frac{1}{v_{2}}-\frac{1}{u_{2}}=\frac{1}{f_{2}}\)
where,
u₂ = Object distance = +(120 —8)=112 cm
v₂= Image distance =?

Magnification, m’ = \(\left|\frac{v_{2}}{u_{2}}\right|=\frac{2240}{92} \times \frac{1}{112}=\frac{20}{92}\)
Hence, the magnification due to the concave lens is \(\frac{20}{92}\)
The magnification produced by the combination of the two lenses is calculated as m x m’ = \(3 \times \frac{20}{92}=\frac{60}{92}\) = 0.652
The magnification of the combination is given as
\(\frac{h_{2}}{h_{1}}\) = 0.652
h₂ = 0.652 x h₁
where, h₁ = Object size = 1.5 cm,
h₂ = Size of the image
∴ h₂ = 0.652 x 1.5 = 0.98 cm
Hence, the height of the image is 0.98 cm.

Question 22.
At what angle should a ray of light be incident on the face of a prism of refracting angle 60° so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is 1.524.
Answer:
The incident, refracted, and emergent rays associated with a glass prism ABC are shown in the given figure

Angle of prism, A = 60°
Refractive index of the prism, μ = 1.524
i₁ = Incident angle
r₂ = Refracted angle
r₂ = Angle of incidence at the face
AC = e = Emergent angle = 90°

According to Snell’s law, for face AC, we can have sine
\(\frac{\sin e}{\sin r_{2}}\) = μ

It is clear from the figure that angle A = r₁ + r₂
According to Snell’s law, we have the relation
μ = \(\frac{\sin i_{1}}{\sin r_{1}} \)
sin i₁ = μ sin r₁
= 1.524 x sin19°= 0.496
∴ i₁= 29.75°
Hence, the angle of incidence is 29.75°.

Question 23.
You are given prisms made of crown glass and flint glass with a wide variety of angles. Suggest a combination of prisms which will
(a) deviate a pencil of white light without much dispersion,
(b) disperse (and displace) a pencil of white light without much deviation.
Answer:
(a) Place the two prisms beside each other. Make sure that their bases are on the opposite sides of the incident white light, with their faces touching each other. When the white light is incident on the first prism, it will get dispersed. When this dispersed light is incident on the second prism, it will recombine and white light will emerge from the combination of the two prisms.

(b) Take the system of the two prisms as suggested in answer (a). Adjust (increase) the angle of the flint-glass-prism so that the deviations due to the combination of the prisms become equal. This combination will disperse the pencil of white light without much deviation.

Question 24.
For a normal eye, the far point is at infinity and the near point of distinct vision is about 25 cm in front of the eye. The cornea of . the eye provides a converging power of about 40 dioptres, and the least converging power of the eye-lens behind the cornea is about 20 dioptres. From this rough data estimate the range of accommodation (i.e., the range of converging power of the eye-lens) of a normal eye.
Answer:
Least distance of distinct vision, d = 25 cm
Far point of a normal eye, d’ = ∞
Converging power of the cornea, P_c = 40 D
Least converging power of the eye- lens, P_e = 20 D
To see the objects at infinity, the eye uses its least converging power.
Power of the eye-lens, P = P_c + P_e =40+20 = 60 D
Power of the eye-lens is given as
P = \(\frac{1}{\text { Focal length of the eye lens }(f)} \)
f = \(=\frac{1}{P}=\frac{1}{60 D}=\frac{100}{60}=\frac{5}{3}\) cm

To focus an object at the near point, object distance (u) = -d = -25 cm
Focal length of the eye-lens = Distance between the cornea and the retina = Image distance
Hence, image distance, v = \( \frac{5}{3}\) cm
According to the lens formula, we can write
\(\frac{1}{f^{\prime}}=\frac{1}{v}-\frac{1}{u}\)
Where f’ = Focal length

Power of the eye-lens = 64-40 = 24 D
Hence, the range of accommodation of the eye-lens is from 20 D to24D.

Question 25.
Does short-sightedness (myopia) or long-sightedness (hypermetropia) imply necessarily that the eye has partially lost its ability of accommodation? If not, what might cause these defects of vision?
Answer:
A myopic or hypermetropic person can also possess the normal ability of accommodation of the eye-lens. Myopia occurs when the eyeballs get elongated from front to back. Hypermetropia occurs when the eye-balls get shortened.
When the eye-lens loses its ability of accommodation, the defect is called presbyopia.

Question 26.
A myopic person has been using spectacles of power -1.0 dioptre for distant vision. During old age he also needs to use separate reading glass of power +2.0 dioptres. Explain what may have happened.
Answer:
The power of the spectacles used by the myopic person, P = -1.0 D
Focal length of the spectacles, f = \(\frac{1}{P}=\frac{1}{-1 \times 10^{-2}}\) = -100 cm
Hence, the far point of the person is 100 cm. He might have a normal near point of 25 cm. When he uses the spectacles, the objects placed at infinity produce virtual images at 100 cm.
He uses the ability of accommodation of the eye-lens to see the objects placed between 100 cm and 25 cm.
During old age, the person uses reading glasses of power, P’ = +2D The ability of accommodation is lost in old age.
This defect is called presbyopia. As a result, he is unable to see clearly the objects placed at 25 cm.

Question 27.
A person looking at a person wearing a shirt with a pattern comprising vertical and horizontal lines is able to see the vertical lines more distinctly than the horizontal ones. What is this defect due to? How is such a defect of vision corrected?
Answer:
In the given case, the person is able to see vertical lines more distinctly than horizontal lines. This means that the refracting system (cornea and eye-lens) of the eye is not working in the same way in different planes. This defect is tailed astigmatism. The person’s eye has enough curvature in the vertical plane. However, the curvature in the horizontal plane is insufficient. Hence, sharp images of the vertical lines are formed on the retina, but horizontal lines appear blurred. This defect can be corrected by using cylindrical lenses.

Question 28.
A man with normal near point (25cm) reads a book with small print using a magnifying glass: a thin convex lens of focal length 5 cm.
(a) What is the closest and the farthest distance at which he should keep the lens from the page so that he can read the book when viewing through the magnifying glass?
(b) What is the maximum and the minimum angular magnification (magnifying power) possible using the above simple microscope?
Answer:
(a) Focal length of the magnifying glass, f = 5 cm
Least distance of distinct vision, d = 25 cm
Closest object distance = u
Image distance, v = -d = -25 cm
According to the lens formula, we have


Hence, the closest distance at which the person can read the book is 4.167 cm.
For the object at the farthest distance (u’), the image distance (v’) = ∞

According to the lens formula, we have

∴ u’ = -5 cm
Hence, the farthest distance at which the person can read the book is 5 cm.
(b) Maximum angular magnification is given by the relation
α_max= \(\frac{d}{|u|}=\frac{25}{\frac{25}{6}} \) = 6
Minimum angular magnification is given by the relation
α_min = \(\frac{d}{\left|u^{\prime}\right|}=\frac{25}{5} \) = 5.

Question 29.
A card sheet divided into squares each of size 1 mm² is being viewed at a distance of 9 cm through a magnifying glass (a converging lens of focal length 10 cm) held close to the eye.
(a) What is the magnification produced by the lens? How much is the area of each square in the virtual image?
(b) What is the angular magnification (magnifying power) of the lens?
(c) Is the magnification in (a) equal to the magnifying power in (b)?
Explain.
Answer:
(a) Area of each square, A = 1 mm²
Object distance, u = -9 cm
Focal length of the converging lens, f = 10 cm
For image distance v, the lens formula can be written as

∴ v = -90 cm
Magnification, m = \(=\frac{v}{u}=\frac{-90}{-9}\) =10
∴ Area of each square in the virtual image = (10)²A
= 10² x 1 =100 mm² = 1 cm²
(b) Magnifying power of the lens = \(\frac{d}{|u|}=\frac{25}{9}\) = 2.8
(c) The magnification in (a) is not the same as the magnifying power in(b).
The magnification magnitude is \(\left(\left|\frac{v}{u}\right|\right)\) and the magnifying power is \(\left(\frac{d}{|u|}\right) \) .
The two quantities will be equal when the image is formed at the near point (25 cm).

Question 30.
(a) At what distance should the lens be held from the figure in Exercise 9.29 in order to view the squares distinctly with the maximum possible magnifying power?
(b) What is the magnification in this case?
(c) Is the magnification equal to the magnifying power in this case? Explain.
Answer:
(a) The maximum possible magnification is obtained when the image is formed at the near point (d = 25cm).
Image distance, v = -d = -25 cm
Focal length, f = 10 cm
Object distance = u
According to the lens formula, we have

∴ u = \(-\frac{50}{7}\) = -7.14 cm
Hence, to view the squares distinctly, the, lens should be kept 7.14 cm away from them. .
(b) Magnifying = \(\left|\frac{v}{u}\right|=\frac{25}{50}\) =3.5
(c) Magnifying power = \(\frac{d}{u}=\frac{25}{\frac{50}{7}}\) = 3.5
Since the image is formed at the near point (25 cm), the magnifying power is equal to the magnitude of magnification.

Question 31.
What should be the distance between the object in Exercise 9.30 and the magnifying glass if the virtual image of each square in the figure is to have an area of 6.25 mm². Would you be able to see the squares distinctly with your eyes very close to the magnifier? [Note: Exercises 9.29 to 9.31 will help you clearly understand the difference between magnification in absolute size and the angular magnification (or magnifying power) of an instrument.]
Answer:
Area of the virtual image of each square, A = 6.25 mm
Area of each square, A₀ = 1 mm²
Hence, the linear magnification of the object can be calculated as
m = \(\sqrt{\frac{A}{A_{0}}}=\sqrt{\frac{6.25}{1}} \) = 2.5
But m = \(\frac{\text { Image distance }(v)}{\text { Object distance }(u)} \)
∴ v = mu = 2.5 u
Focal length of the magnifying glass, f = 10 cm

According to the lens formula, we have the relation

∴ u = \(-\frac{1.5 \times 10}{2.5}\) = -6 cm
and v = 2.5 u = 2.5 x 6 = -15 cm
The virtual image is formed at a distance of 15 cm, which is less than the near point (i.e., 25 cm) of a normal eye. Hence, it cannot be seen by the eyes distinctly.

Question 32.
Answer the following questions:
(a) The angle subtended at the eye by an object is equal to the angle subtended at the eye by the virtual image produced by a magnifying glass. In what sense then does a magnifying glass provide angular magnification?
(b) In viewing through a magnifying glass, one usually positions one’s eyes very close to the lens. Does angular magnification change if the eye is moved back?
(c) Magnifying power of a simple microscope is inversely proportional to the focal length of the lens. What then stops us from using a convex lens of smaller and smaller focal length and achieving greater and greater magnifying power?
(d) Why must both the objective and the eyepiece of a compound microscope have short focal lengths?
(e) When viewing through a compound microscope, our eyes should be positioned not on the eyepiece but a short distance away from it for best viewing. Why? How much should be that short distance between the eye and eyepiece?
Answer:
(a) Though the image size is bigger than the object, the angular size of the image is equal to the angular size of the object. A magnifying glass helps one see the objects placed closer than the least distance of distinct vision (i.e., 25 cm). A closer object causes a larger angular size. A magnifying glass provides angular magnification. Without magnification, the object cannot be placed closer to the eye. With magnification, the object can be placed much closer to the eye.

(b) Yes, the angular magnification changes when the distance between the eye and a magnifying glass is increased, the angular magnification decreases a little. This is because the angle subtended at the eye is slightly less than the angle subtended at the lens. Image distance does not have any effect on angular magnification.

(c) The focal length of a convex lens cannot be decreased by a greater amount. This is because making lenses having very small focal lengths is not easy. Spherical and chromatic aberrations are produced by a convex lens having a very small focal length.

(d) The ang lar magificarin produced by’the eyepiece of a compound microscope is \(\left[\left(\frac{25}{f_{e}}\right)+1\right]\)
Where f_e = Focal length of the eyepiece
It can be inferred that f_e is small, then angular magnification of the eyepiece will be large.
The angular magnification of the objective lens of a compound microscope is given as
\(\frac{1}{\left(\left|u_{o}\right| f_{o}\right)}\)
Where, u₀ = Object distance for the objective lens, f₀ = Focal length of the objective
The magnification is large when u₀> f₀ . In the case of a microscope, the object is kept close to the objective lens. Hence, the object distance is very little.
Since u₀ is small, f₀ will be even smaller. Therefore, f_e and f₀ are both small in the given condition.

(e) When we place our eyes too close to the eyepiece of a compound microscope, we are unable to collect much-refracted light. As a result, the field of view decreases substantially. Hence, the clarity of the image gets blurred.
The best position of the eye for viewing through a compound microscope is at the eye-ring attached to the eyepiece. The precise location of the eye depends on the separation between the objective lens and the eyepiece.

Question 33.
An angular magnification (magnifying power) of 30X is desired using an objective of focal length 1.25 cm and an eyepiece of focal length 5 cm. How will you set up the compound microscope?
Answer:
Focal length of the objective lens, f₀ = 1.25 cm
Focal length of the eyepiece, f_e = 5 cm
Least distance of distinct vision, d = 25 cm

Angular magnification of the compound microscope = 30X
Total magnifying power of the compound microscope, m = 30
The angular magnification of the eyepiece is given by the relation
m_e = \(\left(1+\frac{d}{f_{e}}\right)=\left(1+\frac{25}{5}\right)\) = 1+5 = 6
The angular magnification of the objective lens (m₀) is related to me as
m_om_e=m
or m₀ = \(\frac{m}{m_{e}}=\frac{30}{6}\) = 5

We also have the relation
m = \( \frac{\text { Image distance for the objective lens }\left(v_{o}\right)}{\text { Object distance for the objective lens }\left(-u_{0}\right)}\)
5 = \(\frac{v_{o}}{-u_{o}}\)
∴ v₀ = -5u₀ …………………………….. (1)
Applying the lens formula for the objective lens

and v₀ = -5u₀
= -5 x (-1.5) = 7.5 cm
The object should be placed 1.5 cm away from the objective lens to obtain the desired magnification.
Applying the lens formula for the eyepiece
\(\frac{1}{v_{e}}-\frac{1}{u_{e}}=\frac{1}{f_{e}}\)
where,
v_e = Image distance for the eyepiece = -d = -25 cm
u_e = Object distance for the eyepiece
\(\frac{1}{u_{e}}=\frac{1}{v_{e}}-\frac{1}{f_{e}}=\frac{-1}{25}-\frac{1}{5}=-\frac{6}{25}\)
u_e =-4.17 cm
Separation between the objective lens and the eyepiece = \(\left|u_{e}\right|+\left|v_{o}\right|\)
= 4.17 + 7.5 = 11.67 cm
Therefore, the separation between the objective lens and the eyepiece should be 11.67 cm.

Question 34.
A small telescope has an objective lens of focal length 140 cm and an eyepiece of focal length 5.0 cm. What is the magnifying power of the telescope for viewing distant objects when
(a) the telescope is in normal adjustment (i.e., when the final image is at infinity)?
(b) the final image is formed at the least distance of distinct vision (25 cm)?
Answer:
Focal length of the objective lens, f₀ =140 cm
Focal length of the eyepiece, f_e = 5 cm
Least distance of distinct vision, d = 25 cm
(a) When the telescope is in normal adjustment, its magnifying power is given as
m = \(\frac{f_{o}}{f_{e}}=\frac{140}{5} \) = 28
(b) When the final image is formed at d, the magnifying power of the telescope is given as
\(\frac{f_{o}}{f_{e}}\left[1+\frac{f_{e}}{d}\right]=\frac{140}{5}\left[1+\frac{5}{25}\right]\)
= 28[1 +0.2] = 28×1.2 = 33.6

Question 35.
(a) For the telescope described in Exercise 9.34 (a), what is the separation between the objective lens and the eyepiece?
(b) If this telescope is used to view a 100 m tall tower 3 km away, what is the height of the image of the tower formed by the objective lens?
(c) What is the height of the final image of the tower if it is formed at 25 cm?
Answer:
Focal length of the objective lens, f₀ =140 cm
Focal length of the eyepiece, f_e= 5 cm
(a) In normal adjustment, the separation between the objective lens and the eyepiece = f₀ + f_e = 140 + 5 = 145 cm
(b) Height of the tower, h₁ = 100 m
Distance of the tower (object) from the telescope, u = 3 km = 3000 m
The angle subtended by the tower at the telescope is given as
θ’ = \(\frac{h_{2}}{f_{o}}=\frac{h_{2}}{140}\) rad
where,
h₂ = Height of the image of the tower formed by the objective lens
\(\frac{1}{30}=\frac{h_{2}}{140}\) (∵θ=θ’)
∴ h₂ = \(\frac{140}{30}\) = 4.7 cm
Therefore, the objective lens forms a 4.7 cm tall image of the tower.

(c) Image is formed at a distance, d = 25 cm
The magnification of the eyepiece is given by the relation
m = 1 + \(\frac{d}{f_{e}}\)
= 1+ \(\frac{25}{5}\) =1 + 5 = 6
Height of the final image = mh₂ = 6 x 4.7 = 28.2 cm
Hence, the height of the final image of the tower is 28.2 cm.

Question 36.
A Cassegrain telescope uses two mirrors as shown in Fig. 9.33. Such a telescope is built with the mirrors 20 mm apart.
If the radius of curvature of the large mirror is 220 mm and the small mirror is 140 mm, where will the final image of an object at infinity be?
Answer:
Given, r₁ = 220 mm, f₁ = \(\frac{r_{1}}{2}\) = 110 mm = 11 cm
r₂ = 140 mm, f₂ = \(\frac{r_{2}}{2}\) = 70 mm = 7.0 cm
Distance between mirrors, d = 20 mm = 2.0 cm
The parallel incident rays coming from distant objects fall on the concave mirror and try to be focused at the principal focus of concave lens, i. e., v₁ = -f₁ = -11 cm
But in the path of rays reflected from concave mirror, a convex mirror is placed. Therefore the image formed by the concave mirror acts as a virtual object for convex mirror.

For convex mirror f₂ = -7.0 cm, u₂ = -(11 -2) = -9 cm

v₂ = \(-\frac{63}{2}\) cm = -31.5 cm
This is the distance of the final image formed by the convex mirror. Thus, the final image is formed at a distance of 31.5 cm from the smaller (convex) mirror behind the bigger mirror.

Question 37.
Light incident normally on a plane mirror attached to a galvanometer coil retraces backward as shown in Fig. 9.36. A current in the coil produces a deflection of 3.5° of the mirror. What is the displacement of the reflected spot of light on a screen placed 1.5 m away?

Answer:
Angle of deflection, θ = 3.5°
Distance of the screen from the mirror, D = 1.5 m
The reflected rays get deflected by an amount twice the angle of deflection i.e., 2θ = 2 x 3.5 = 7.0°
The displacement (d) of the reflected spot of light on the screen is given as
tan 2θ = \(\frac{d}{1.5}\) d =1.5 x tan7°= 0.184 m = 18.4 cm
Hence, the displacement of the reflected spot of light is 18.4 cm.

Question 38.
Figure 9.37 shows an biconvex lens (of refractive index 1.50) in contact with a liquid layer on top of a plane mirror.
A small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be 45.0 cm. The liquid is removed and the experiment is repeated. The new distance is measured to he 30.0 cm. What is the refractive index of the liquid?

Answer:
Focal length of the convex lens, f₁ = 30 cm
The liquid acts as a mirror. Focal length of the liquid = f₂
Focal length of the system (convex lens + liquid), f = 45 cm
For a pair of optical systems placed in contact, the equivalent focal length is given as
\(\frac{1}{f}=\frac{1}{f_{1}}+\frac{1}{f_{2}}\)
\(\frac{1}{f_{2}}=\frac{1}{f}-\frac{1}{f_{1}}\)
= \(\frac{1}{45}-\frac{1}{30}=-\frac{1}{90}\)
∴ f₂ = -90 cm
Let the refractive index of the lens be μ₁ and the radius of curvature of one surface be R. Hence, the radius of curvature of the other surface is R.
R can be obtained using the relation \(\frac{1}{f_{1}}=\left(\mu_{1}-1\right)\left(\frac{1}{R}+\frac{1}{-R}\right)\)
\(\frac{1}{30}=(1.5-1)\left(\frac{2}{R}\right)\)
∴ R = \(\frac{30}{0.5 \times 2}\) = 30 cm

Let μ₂ be the refractive index of the liquid.
Radius of curvature of the liquid on the side of the plane minor = ∞
Radius of curvature of the liquid on the side of the lens, R = -30 cm
The value of μ₂, can be calculated using the relation
\(\frac{1}{f_{2}}=\left(\mu_{2}-1\right)\left[\frac{1}{-R}-\frac{1}{\infty}\right]\)
\(\frac{-1}{90}=\left(\mu_{2}-1\right)\left[\frac{1}{+30}-0\right]\)
μ₂ – 1 = \(\frac{1}{3} \)
∴ μ₂ = \(\frac{4}{3} \) = 133
Hence, the refractive index of the liquid is 1.33.