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.]
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 .5 cm
Thus, the wavelength of the electromagnetic wave is 1.5 cm .
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 = c B = 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.
Given: The electric field amplitude of an electromagnetic wave is
a)
The wavelength of the wave is given as,
Where, the wavelength is
By substituting the given values in the above expression, we get
Thus, the wavelength of the electromagnetic wave is
b)
The magnetic field strength is given as,
Where, the electric field strength is
By substituting the given values in the above expression, we get
Thus, the magnetic field strength is
c)
The energy density of the electric field is given as,
Where, the permittivity of free space is
The energy density of the magnetic field is given as,
Where, the permeability of free space is
The electric field and magnetic field in free space are related as,
By substituting the value of
From equation (2) and (4), we get
Thus, the energy density of the electric field is equal to the energy density of the magnetic field.
