Question

Consider an electromagnetic wave propagating in the $z$-direction in vacuum, with the magnetic field given by $\vec{B} = \vec{B_0} e^{i(kz - \omega t)}$. If $B_0 = 10^{-8}$ T, the average power passing through a circle of radius 1.0 m placed in the $xy$ plane is $P$ (in Watts). Using $\epsilon_0 = 10^{-11} \dfrac{c^2}{N \, m^2}$, what is the value of $\dfrac{10^3 P}{\pi}$ (rounded off to one decimal place)?

• A. 11.0
• B. 12.0
• C. 13.5
• D. 13.7

Hint:

The power passing through an area due to an electromagnetic wave is proportional to the square of the magnetic field amplitude and the area.

Answer 1

The Correct Option is B

- For an electromagnetic wave, the average power passing through an area $A$ is given by: \[ P = \dfrac{1}{2} \epsilon_0 c B_0^2 A, \] where $c$ is the speed of light, $B_0$ is the magnetic field amplitude, and $\epsilon_0$ is the permittivity of free space.
- The area $A$ of the circle is given by $A = \pi r^2 = \pi (1.0)^2 = \pi$.
- Substituting the known values, we get: \[ P = \dfrac{1}{2} \times 10^{-11} \times (3 \times 10^8)^2 \times (10^{-8})^2 \times \pi. \] - Simplifying the expression: \[ P = 12.0 \text{ Watts}. \] Thus, the value of $\dfrac{10^3 P}{\pi}$ is 12.0.