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Question

In a plane electromagnetic wave, the electric field oscillatessinusoidally 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 theaverage energy density of the B field. [c = 3 × 108 m s–1.]

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Solution

Given: The electric field amplitude of an electromagnetic wave is 48 Vm 1 and the frequency of the source is 2.0× 10 10 Hz.

a)

The wavelength of the wave is given as,

λ= c ν

Where, the wavelength is λ, the speed of light is c and the frequency is ν.

By substituting the given values in the above expression, we get

λ= 3× 10 8 2× 10 10 =1.5× 10 2 m =1.5cm

Thus, the wavelength of the electromagnetic wave is 1.5cm.

b)

The magnetic field strength is given as,

B 0 = E 0 c

Where, the electric field strength is E 0 and the speed of the light is c.

By substituting the given values in the above expression, we get

B 0 = 48 3× 10 8 =1.6× 10 7 T

Thus, the magnetic field strength is 1.6× 10 7 T.

c)

The energy density of the electric field is given as,

U E = 1 2 ε 0 E 2 (1)

Where, the permittivity of free space is ε 0 and the electric field is E.

The energy density of the magnetic field is given as,

U B = B 2 2 μ 0 (2)

Where, the permeability of free space is μ 0 and the magnetic field is B.

The electric field and magnetic field in free space are related as,

E=cB = 1 ε 0 μ 0 B (3)

By substituting the value of E from the equation (3) in the equation (1), we get

U E = 1 2 ε 0 ( B ε 0 μ 0 ) 2 = 1 2 μ 0 B 2 (4)

From equation (2) and (4), we get

U E = U B

Thus, the energy density of the electric field is equal to the energy density of the magnetic field.


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