Question

A plane electromagnetic wave is passing through a region. Consider (a) electric field (b) magnetic field (c) electrical energy in a small volume and (d) magnetic energy in a small volume. Construct the pairs of the quantities that oscillate with equal frequencies.

Answer 1

Let the electromagnetic wave be propagating in the z-direction. The vibrations of the electric and magnetic fields are given by:
E_x= E₀ sin (kz – ωt)
B_y= B₀ sin (kz – ωt)
Let the volume of the region be V.
The angular frequency of the vibrations of the electric and magnetic fields are same and are equal to ω. Therefore, their frequency, `f = ω/(2pi)` , is same .

The electrical energy in the region,

`U_E = (1/2 ∈_0 E^2) xx V`

It can be written as :

`U_E = (1/2 ∈_0 (E_0^2 sin^2(kz - ωt))xx V`

`U_E = (1/2 ∈_0 E_0^2 xx ((1-cos2(kz-ωt)))/2) xx V`

`U_E = (1/4 ∈_0 E_0^2 xx (1-cos2(kz-ωt))) xx V`

The magnetic energy in the region ,

`U_B = (B^2/(2u_0)) xx V`

`U_B = ((B_0^2 sin^2(kz-ωt))/(2u_0)) xx V`

⇒ `U_B = ((B_0^2(1-cos(2kz-2ωt)))/(4u_0)) xx V`

The angular frequency of the electric and magnetic energies is same and is equal to 2ω.Therefore, their frequency,

`f^' = (2ω)/(2pi) = 2f`

, will be same.

Thus, the electric and magnetic fields have same frequencies and the electrical and magnetic energies will have same frequencies.