Class XII Chapter 8 – Electromagnetic Waves Physics

∴B

= 1.63 × 10−11 T

Hence, the magnetic field at that point is 1.63 × 10−11 T.

Question 8.3:

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

The speed of light (3 × 108 m/s) in a vacuum is the same for all wavelengths. It is

independent of the wavelength in the vacuum.

Question 8.4:

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

The electromagnetic wave travels in a vacuum along the z-direction. The electric field (E)

and the magnetic field (H) are in the x-y plane. They are mutually perpendicular.

Frequency of the wave, ν = 30 MHz = 30 × 106 s−1

Speed of light in a vacuum, c = 3 × 108 m/s

Wavelength of a wave is given as:

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Question 8.5:

A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding

wavelength band?

Answer

A radio can tune to minimum frequency, ν1 = 7.5 MHz= 7.5 × 106 Hz

Maximum frequency, ν2 = 12 MHz = 12 × 106 Hz

Speed of light, c = 3 × 108 m/s

Corresponding wavelength for ν1 can be calculated as:

Corresponding wavelength for ν2 can be calculated as:

Thus, the wavelength band of the radio is 40 m to 25 m.

Question 8.6:

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

The frequency of an electromagnetic wave produced by the oscillator is the same as that of

a charged particle oscillating about its mean position i.e., 109 Hz.

Question 8.7:

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?

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Answer

Amplitude of magnetic field of an electromagnetic wave in a vacuum,

B0 = 510 nT = 510 × 10−9 T

Speed of light in a vacuum, c = 3 × 108 m/s

Amplitude of electric field of the electromagnetic wave is given by the relation,

E = cB0

= 3 × 108 × 510 × 10−9 = 153 N/C

Therefore, the electric field part of the wave is 153 N/C.

Question 8.8:

Suppose that the electric field amplitude of an electromagnetic wave is E0 = 120 N/C and

that its frequency is ν = 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, ν = 50.0 MHz = 50 × 106 Hz

Speed of light, c = 3 × 108 m/s

(a) Magnitude of magnetic field strength is given as:

Angular frequency of source is given as:

ω = 2πν

= 2π × 50 × 106

= 3.14 × 108 rad/s

Propagation constant is given as:

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Wavelength of wave is given as:

(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:

And, magnetic field vector is given as:

Question 8.9:

The terminology of different parts of the electromagnetic spectrum is given in the text. Use

the formula E = hν (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:

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Where,

h = Planck’s constant = 6.6 × 10−34 Js

c = Speed of light = 3 × 108 m/s

λ = Wavelength of radiation

The given table lists the photon energies for different parts of an electromagnetic spectrum

for differentλ.

λ 103 1 10−3 10−6 10−8 10−10 10−12

(m)

E 12.375 × 12.375 × 12.375 × 12.375 × 12.375 × 12.375 × 12.375 ×

(eV) 10−10 10−7 10−4 10−1 101 103 105

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 8.10:

In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of

2.0 × 1010 Hz and amplitude 48 V m−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 m s−1.]

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Answer

Frequency of the electromagnetic wave, ν = 2.0 × 1010 Hz

Electric field amplitude, E0 = 48 V m−1

Speed of light, c = 3 × 108 m/s

(a) Wavelength of a wave is given as:

(b) Magnetic field strength is given as:

(c) Energy density of the electric field is given as:

And, energy density of the magnetic field is given as:

Where,

∈0 = Permittivity of free space

µ0 = Permeability of free space

We have the relation connecting E and B as:

E = cB … (1)

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Where,

… (2)

Putting equation (2) in equation (1), we get

Squaring both sides, we get

Question 8.11:

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]} .

(a) What is the direction of propagation?

(b) What is the wavelength λ?

(c) What is the frequency ν?

(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) From the given electric field vector, it can be inferred that the electric field is directed

along the negative x direction. Hence, the direction of motion is along the negative y

direction i.e., .

(b) It is given that,

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The general equation for the electric field vector in the positive x direction can be written

as:

On comparing equations (1) and (2), we get

Electric field amplitude, E0 = 3.1 N/C

Angular frequency, ω = 5.4 × 108 rad/s

Wave number, k = 1.8 rad/m

Wavelength, = 3.490 m

(c) Frequency of wave is given as:

(d) Magnetic field strength is given as:

Where,

c = Speed of light = 3 × 108 m/s

(e) On observing the given vector field, it can be observed that the magnetic field vector is

directed along the negative z direction. Hence, the general equation for the magnetic field

vector is written as:

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Question 8.12:

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 rating of bulb, P = 100 W

It is given that about 5% of its power is converted into visible radiation.

Power of visible radiation,

Hence, the power of visible radiation is 5W.

(a) Distance of a point from the bulb, d = 1 m

Hence, intensity of radiation at that point is given as:

(b) Distance of a point from the bulb, d1 = 10 m

Hence, intensity of radiation at that point is given as:

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Question 8.13:

Use the formula λm 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 continous 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,

Where,

λm = maximum wavelength

T = temperature

Thus, the temperature for different wavelengths can be obtained as:

For λm = 10−4 cm;

For λm = 5 ×10−5 cm;

For λm = 10−6 cm; 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 8.14:

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).

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(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 57Fe nucleus associated with a famous high

resolution spectroscopic method

(Mössbauer spectroscopy)].

Answer

(a) Radio waves; it belongs to the short wavelength end of the electromagnetic spectrum.

(b) Radio waves; it belongs to the short wavelength end.

(c) Temperature, T = 2.7 °K

λm is given by Planck’s law as:

This wavelength corresponds to microwaves.

(d) This is the yellow light of the visible spectrum.

(e) Transition energy is given by the relation,

E = hν

Where,

h = Planck’s constant = 6.6 × 10−34 Js

ν = Frequency of radiation

Energy, E = 14.4 K eV

This corresponds to X-rays.

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Question 8.15:

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 radio telescopes 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 shortwave bands because only these bands can be

refracted by the ionosphere.

(b) 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 radio waves can penetrate it. Hence, optical and radio telescopes 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 atmosphere 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 theabsenceof 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

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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.

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Class XII Chapter 8 – Electromagnetic Waves Physics

Question 8.1:

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 Kirchhoff’s first rule (junction rule) valid at each plate of the capacitor? Explain.

Answer

Radius of each circular plate, r = 12 cm = 0.12 m

Distance between the plates, d = 5 cm = 0.05 m

Charging current, I = 0.15 A

Permittivity of free space, = 8.85 × 10−12 C2 N−1 m−2

(a) Capacitance between the two plates is given by the relation,

C

Where,

A = Area of each plate

Charge on each plate, q = CV

Where,

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V = Potential difference across the plates

Differentiation on both sides with respect to time (t) gives:

Therefore, the change in potential difference between the plates is 1.87 ×109 V/s.

(b) The displacement current across the plates is the same as the conduction current.

Hence, the displacement current, id is 0.15 A.

(c) Yes

Kirchhoff’s first rule is valid at each plate of the capacitor provided that we take the sum of

conduction and displacement for current.

Question 8.2:

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.

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Answer

Radius of each circular plate, R = 6.0 cm = 0.06 m

Capacitance of a parallel plate capacitor, C = 100 pF = 100 × 10−12 F

Supply voltage, V = 230 V

Angular frequency, ω = 300 rad s−1

(a) Rms value of conduction current, I

Where,

XC = Capacitive reactance

∴ I = V × ωC

= 230 × 300 × 100 × 10−12

= 6.9 × 10−6 A

= 6.9 µA

Hence, the rms value of conduction current is 6.9 µA.

(b) Yes, conduction current is equal to displacement current.

(c) Magnetic field is given as:

B

Where,

µ0 = Free space permeability

I0 = Maximum value of current =

r = Distance between the plates from the axis = 3.0 cm = 0.03 m

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∴B

= 1.63 × 10−11 T

Hence, the magnetic field at that point is 1.63 × 10−11 T.

Question 8.3:

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

The speed of light (3 × 108 m/s) in a vacuum is the same for all wavelengths. It is

independent of the wavelength in the vacuum.

Question 8.4:

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

The electromagnetic wave travels in a vacuum along the z-direction. The electric field (E)

and the magnetic field (H) are in the x-y plane. They are mutually perpendicular.

Frequency of the wave, ν = 30 MHz = 30 × 106 s−1

Speed of light in a vacuum, c = 3 × 108 m/s

Wavelength of a wave is given as:

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Question 8.5:

A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding

wavelength band?

Answer

A radio can tune to minimum frequency, ν1 = 7.5 MHz= 7.5 × 106 Hz

Maximum frequency, ν2 = 12 MHz = 12 × 106 Hz

Speed of light, c = 3 × 108 m/s

Corresponding wavelength for ν1 can be calculated as:

Corresponding wavelength for ν2 can be calculated as:

Thus, the wavelength band of the radio is 40 m to 25 m.

Question 8.6:

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

The frequency of an electromagnetic wave produced by the oscillator is the same as that of

a charged particle oscillating about its mean position i.e., 109 Hz.

Question 8.7:

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?

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Answer

Amplitude of magnetic field of an electromagnetic wave in a vacuum,

B0 = 510 nT = 510 × 10−9 T

Speed of light in a vacuum, c = 3 × 108 m/s

Amplitude of electric field of the electromagnetic wave is given by the relation,

E = cB0

= 3 × 108 × 510 × 10−9 = 153 N/C

Therefore, the electric field part of the wave is 153 N/C.

Question 8.8:

Suppose that the electric field amplitude of an electromagnetic wave is E0 = 120 N/C and

that its frequency is ν = 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, ν = 50.0 MHz = 50 × 106 Hz

Speed of light, c = 3 × 108 m/s

(a) Magnitude of magnetic field strength is given as:

Angular frequency of source is given as:

ω = 2πν

= 2π × 50 × 106

= 3.14 × 108 rad/s

Propagation constant is given as:

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Wavelength of wave is given as:

(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:

And, magnetic field vector is given as:

Question 8.9:

The terminology of different parts of the electromagnetic spectrum is given in the text. Use

the formula E = hν (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:

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Where,

h = Planck’s constant = 6.6 × 10−34 Js

c = Speed of light = 3 × 108 m/s

λ = Wavelength of radiation

The given table lists the photon energies for different parts of an electromagnetic spectrum

for differentλ.

λ 103 1 10−3 10−6 10−8 10−10 10−12

(m)

E 12.375 × 12.375 × 12.375 × 12.375 × 12.375 × 12.375 × 12.375 ×

(eV) 10−10 10−7 10−4 10−1 101 103 105

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 8.10:

In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of

2.0 × 1010 Hz and amplitude 48 V m−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 m s−1.]

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Answer

Frequency of the electromagnetic wave, ν = 2.0 × 1010 Hz

Electric field amplitude, E0 = 48 V m−1

Speed of light, c = 3 × 108 m/s

(a) Wavelength of a wave is given as:

(b) Magnetic field strength is given as:

(c) Energy density of the electric field is given as:

And, energy density of the magnetic field is given as:

Where,

∈0 = Permittivity of free space

µ0 = Permeability of free space

We have the relation connecting E and B as:

E = cB … (1)

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Where,

… (2)

Putting equation (2) in equation (1), we get

Squaring both sides, we get

Question 8.11:

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]} .

(a) What is the direction of propagation?

(b) What is the wavelength λ?

(c) What is the frequency ν?

(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) From the given electric field vector, it can be inferred that the electric field is directed

along the negative x direction. Hence, the direction of motion is along the negative y

direction i.e., .

(b) It is given that,

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The general equation for the electric field vector in the positive x direction can be written

as:

On comparing equations (1) and (2), we get

Electric field amplitude, E0 = 3.1 N/C

Angular frequency, ω = 5.4 × 108 rad/s

Wave number, k = 1.8 rad/m

Wavelength, = 3.490 m

(c) Frequency of wave is given as:

(d) Magnetic field strength is given as:

Where,

c = Speed of light = 3 × 108 m/s

(e) On observing the given vector field, it can be observed that the magnetic field vector is

directed along the negative z direction. Hence, the general equation for the magnetic field

vector is written as:

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Question 8.12:

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 rating of bulb, P = 100 W

It is given that about 5% of its power is converted into visible radiation.

Power of visible radiation,

Hence, the power of visible radiation is 5W.

(a) Distance of a point from the bulb, d = 1 m

Hence, intensity of radiation at that point is given as:

(b) Distance of a point from the bulb, d1 = 10 m

Hence, intensity of radiation at that point is given as:

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Question 8.13:

Use the formula λm 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 continous 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,

Where,

λm = maximum wavelength

T = temperature

Thus, the temperature for different wavelengths can be obtained as:

For λm = 10−4 cm;

For λm = 5 ×10−5 cm;

For λm = 10−6 cm; 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 8.14:

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).

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(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 57Fe nucleus associated with a famous high

resolution spectroscopic method

(Mössbauer spectroscopy)].

Answer

(a) Radio waves; it belongs to the short wavelength end of the electromagnetic spectrum.

(b) Radio waves; it belongs to the short wavelength end.

(c) Temperature, T = 2.7 °K

λm is given by Planck’s law as:

This wavelength corresponds to microwaves.

(d) This is the yellow light of the visible spectrum.

(e) Transition energy is given by the relation,

E = hν

Where,

h = Planck’s constant = 6.6 × 10−34 Js

ν = Frequency of radiation

Energy, E = 14.4 K eV

This corresponds to X-rays.

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Question 8.15:

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 radio telescopes 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 shortwave bands because only these bands can be

refracted by the ionosphere.

(b) 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 radio waves can penetrate it. Hence, optical and radio telescopes 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 atmosphere 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 theabsenceof 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

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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.

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Question 9.1:

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,

Image distance = v

The image distance can be obtained using the mirror formula:

Therefore, the screen should be placed 54 cm away from the mirror to obtain a sharp

image.

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 9.2:

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

Height of the needle, h1 = 4.5 cm

Object distance, u = −12 cm

Focal length of the convex mirror, f = 15 cm

Image distance = v

The value of v can be obtained using the mirror formula:

Hence, the image of the needle is 6.7 cm away from the mirror. Also, it is on the other

side of the mirror.

The image size is given by the magnification formula:

Hence, magnification of the image,

The height of the image is 2.5 cm. The positive sign indicates that the image is erect,

virtual, and diminished.

If the needle is moved farther from the mirror, the image will also move away from the

mirror, and the size of the image will reduce gradually.

Question 9.3:

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

Actual depth of the needle in water, h1 = 12.5 cm

Apparent depth of the needle in water, h2 = 9.4 cm

Refractive index of water = μ

The value of μ can be obtained as follows:

Hence, the refractive index of water is about 1.33.

Water is replaced by a liquid of refractive index,

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:

Hence, the new apparent depth of the needle is 7.67 cm. 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 9.4:

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 45º 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:

As per the given figure, for the air − water interface:

Angle of incidence, i = 60°

Angle of refraction, r = 47°

The relative refractive index of water with respect to air is given by Snell’s law as:

Using (1) and (2), the relative refractive index of glass with respect to water can be

obtained as:

The following figure shows the situation involving the glass − water interface.

Angle of incidence, i = 45°

Angle of refraction = r

From Snell’s law, r can be calculated as:

Hence, the angle of refraction at the water − glass interface is 38.68°.

Question 9.5:

A small bulb is placed at the bottom of a tank containing 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,

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,

Using Snell’ law, we can write the relation for the refractive index of water as:

Using the given figure, we have the relation:

∴R = tan 48.75° × 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 9.6:

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, = 40° as:

Angle of the prism, A = 60°

Refractive index of water, µ = 1.33

Refractive index of the material of the prism =

The angle of deviation is related to refractive index

Hence, the refractive index of the material of the prism is 1.532.

Since the prism is placed in water, let 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 9.7:

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

Refractive index of glass,

Focal length of the double-convex lens, f = 20 cm

Radius of curvature of one face of the lens = R1

Radius of curvature of the other face of the lens = R2

Radius of curvature of the double-convex lens = R

The value of R can be calculated as:

Hence, the radius of curvature of the double-convex lens is 22 cm.

Question 9.8:

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 = +12 cm

Focal length of the convex lens, f = 20 cm

Image distance = v

According to the lens formula, we have the relation:

Hence, the image is formed 7.5 cm away from the lens, toward its right.

Focal length of the concave lens, f = −16 cm

Image distance = v

According to the lens formula, we have the relation:

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

Question 9.9:

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 the object, h1 = 3 cm

Object distance, u = −14 cm

Focal length of the concave lens, f = −21 cm

Image distance = v

According to the lens formula, we have the relation:

Hence, the image is formed on the other side of the lens, 8.4 cm away from it. The

negative sign shows that the image is erect and virtual.

The magnification of the image is given as:

Hence, the height of the image is 1.8 cm.

If the object is moved further away from the lens, then the virtual image will move

toward the focus of the lens, but not beyond it. The size of the image will decrease with

the increase in the object distance.

Question 9.10:

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

Focal length of the convex lens, f1 = 30 cm

Focal length of the concave lens, f2 = −20 cm

Focal length of the system of lenses = f

The equivalent focal length of a system of two lenses in contact is given as:

Hence, the focal length of the combination of lenses is 60 cm. The negative sign indicates

that the system of lenses acts as a diverging lens.

Question 9.11:

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 (25 cm), and (b) at infinity? What is the magnifying power of

the microscope in each case?

Answer

Focal length of the objective lens, f1 = 2.0 cm

Focal length of the eyepiece, f2 = 6.25 cm

Distance between the objective lens and the eyepiece, d = 15 cm

Least distance of distinct vision,

∴Image distance for the eyepiece, v2 = −25 cm

Object distance for the eyepiece = u2

According to the lens formula, we have the relation:

Image distance for the objective lens,

Object distance for the objective lens = u1

According to the lens formula, we have the relation:

Magnitude of the object distance, = 2.5 cm

The magnifying power of a compound microscope is given by the relation:

Hence, the magnifying power of the microscope is 20.

The final image is formed at infinity.

∴Image distance for the eyepiece,

Object distance for the eyepiece = u2

According to the lens formula, we have the relation:

Image distance for the objective lens,

Object distance for the objective lens = u1

According to the lens formula, we have the relation:

Magnitude of the object distance, = 2.59 cm

The magnifying power of a compound microscope is given by the relation:

Hence, the magnifying power of the microscope is 13.51.

Question 9.12:

A person with a normal near point (25 cm) 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, fo = 8 mm = 0.8 cm

Focal length of the eyepiece, fe = 2.5 cm

Object distance for the objective lens, uo = −9.0 mm = −0.9 cm

Least distance of distant vision, d = 25 cm

Image distance for the eyepiece, ve = −d = −25 cm

Object distance for the eyepiece =

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

We can also obtain the value of the image distance for the objective lens using the

lens formula.

The distance between the objective lens and the eyepiece

The magnifying power of the microscope is calculated as:

Hence, the magnifying power of the microscope is 88.

Question 9.13:

A small telescope has an objective lens of focal length 144 cm and an eyepiece of focal

length 6.0 cm. 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, fo = 144 cm

Focal length of the eyepiece, fe = 6.0 cm

The magnifying power of the telescope is given as:

The separation between the objective lens and the eyepiece is calculated as:

Hence, the magnifying power of the telescope is 24 and the separation between the

objective lens and the eyepiece is 150 cm.

Question 9.14:

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?

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 × 106 m, and the radius of

lunar orbit is 3.8 × 108 m.

Answer

Focal length of the objective lens, fo = 15 m = 15 × 102 cm

Focal length of the eyepiece, fe = 1.0 cm

The angular magnification of a telescope is given as:

Hence, the angular magnification of the given refracting telescope is 1500.

Diameter of the moon, d = 3.48 × 106 m

Radius of the lunar orbit, r0 = 3.8 × 108 m

Let be the diameter of the image of the moon formed by the objective lens.

The angle subtended by the diameter of the moon is equal to the angle subtended by the

image.

Hence, the diameter of the moon’s image formed by the objective lens is 13.74 cm

Question 9.15:

Use the mirror equation to deduce that:

an object placed between f and 2f of a concave mirror produces a real image beyond 2f.

a convex mirror always produces a virtual image independent of the location of the

object.

the virtual image produced by a convex mirror is always diminished in size and is located

between the focus and the pole.

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

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

For image distance v, we can write the lens formula as:

The object lies between f and 2f.

Using equation (1), we get:

∴ is negative, i.e., v is negative.

Therefore, the image lies beyond 2f.

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:

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.

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:

Hence, the image formed is diminished and is located between the focus (f) and the pole.

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.

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: