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, $f=\frac{\omega }{2\pi }$, is same.
The electrical energy in the region,
U_E = $\left(\frac{1}{2}{\in }_0{E}^2\right)\times V$
It can be written as:
$$\begin{gathered} U_{\mathrm{E}}=\left(\frac{1}{2}{\in }_0\left({E_0}^2{\sin }^2\right(kz-\omega t)\right)\times V \\ {U}_E=\left(\frac{1}{2}{\in }_0{E_0}^2\times \frac{\left(1-\cos 2(kz-\omega t)\right)}{2}\right)\times V \\ {U}_E=\left(\frac{1}{4}{\in }_0{E_0}^2\times \left(1-\cos 2(kz-\omega t)\right)\right)\times V \end{gathered}$$
The magnetic energy in the region,
$$\begin{gathered} U_{\mathrm{B}}=\left(\frac{B^2}{2{\mu }_0}\right)\times V \\ {U}_{\mathrm{B}}=\left(\frac{{B_0}^2{\sin }^2(kz-\omega t)}{2{\mu }_0}\right)\times V \\ \Rightarrow U_{\mathrm{B}}=\left(\frac{{B_0}^2\left(1-\cos (2kz-2\omega t)\right)}{4{\mu }_0}\right)\times V \end{gathered}$$

The angular frequency of the electric and magnetic energies is same and is equal to 2ω.
Therefore, their frequency, $f\text{'}=\frac{2\omega }{2\pi }=2f$, will be same.

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