GATE 2015 EC – Question 64
The electric field intensity of a plane wave traveling in free space is given by the following expression
$$\mathbf{E}(x, t) = \mathbf{a}_y 24\pi\cos(\omega t - k_0 x) \text{ (V/m)}$$
In this field, consider a square area 10 cm x 10 cm on a plane $x + y = 1$. The total time-averaged power (in mW) passing through the square area is ________.
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Correct answer: 53 to 54
Explanation
The time-averaged power density is $S = \frac{E_0^2}{2\eta_0} = \frac{(24\pi)^2}{2 \times 120\pi} = 2.4\pi = 7.54$ W/m$^2$, directed along $x$. The plane $x + y = 1$ has the normal $\frac{1}{\sqrt{2}}(\hat{x} + \hat{y})$, so the power through the area $0.01$ m$^2$ is $7.54 \times \frac{1}{\sqrt{2}} \times 0.01 = 0.0533$ W, which is 53.3 mW.