Question

Which of the following have zero average value in a plane electromagnetic wave? (A) Electric energy (B) Magnetic energy (C) Electric field (D) None of these.

Short answer

The average value of the electric field in a plane electromagnetic wave is zero (C).

Step-by-step solution

1. Electric Energy

The electric energy density of an electromagnetic wave is given by: \(u_E = \frac{1}{2}\epsilon_0 E^2\) Where \(\epsilon_0\) is the permittivity of free space, and \(E\) is the electric field amplitude. The electric field varies sinusoidally in time, as \(E(t) = E_0 \sin(\omega t)\). To find the average value of electric energy in one time period, we can integrate over time: \(\bar{u}_E = \frac{1}{T}\int_0^{T} \frac{1}{2}\epsilon_0 (E_0 \sin(\omega t))^2 dt \)

2. Magnetic Energy

The magnetic energy density of an electromagnetic wave is given by: \(u_B = \frac{1}{2\mu_0} B^2\) Where \(\mu_0\) is the permeability of free space, and \(B\) is the magnetic field amplitude. The magnetic field also varies sinusoidally in time, as \(B(t) = B_0 \sin(\omega t)\). To find the average value of magnetic energy in one time period, we can integrate over time: \(\bar{u}_B = \frac{1}{T}\int_0^{T} \frac{1}{2\mu_0} (B_0 \sin(\omega t))^2 dt \)

3. Electric field

The electric field in a plane electromagnetic wave varies sinusoidally as \(E(t) = E_0 \sin(\omega t)\). To find the average value of the electric field in one time period, we can integrate over time: \(\bar{E} = \frac{1}{T}\int_0^{T} E_0 \sin(\omega t) dt \)

4. Find the average values of each quantity

For electric energy: \(\bar{u}_E = \frac{1}{T}\int_0^{T} \frac{1}{2}\epsilon_0 (E_0 \sin(\omega t))^2 dt = \frac{1}{2} \epsilon_0 E_0^2 \cdot\frac{1}{2}\) (as the average value of \(\sin^2(\omega t)\) over one period is \(\frac{1}{2}\)) For magnetic energy: \(\bar{u}_B = \frac{1}{T}\int_0^{T} \frac{1}{2\mu_0} (B_0 \sin(\omega t))^2 dt = \frac{1}{2\mu_0} B_0^2 \cdot\frac{1}{2}\) (as the average value of \(\sin^2(\omega t)\) over one period is \(\frac{1}{2}\)) For electric field: \(\bar{E} = \frac{1}{T}\int_0^{T} E_0 \sin(\omega t) dt = E_0 \cdot 0\) (as the average value of \(\sin(\omega t)\) over one period is 0) From the results above, we can see that the electric energy and magnetic energy have non-zero average values in a plane electromagnetic wave, while the average value of the electric field is zero. Therefore, the correct answer is: (C) Electric field