GATE 2023 PH – Question 61
A vacuum wave propagates along z with $\mathbf B=\mathbf B_0e^{i(kz-\omega t)}$ and $B_0=10^{-8}$ T. Average power P crosses a circle of radius 1 m in the xy plane. With $\epsilon_0=10^{-11}\ \mathrm{C^2/(Nm^2)}$, find $10^3P/\pi$ to one decimal place.
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Correct answer: 13.43 to 13.57
Explanation
**Intensity of a plane electromagnetic wave** in vacuum (time-averaged):
$$I=\frac12\epsilon_0c\,E_0^2,\qquad E_0=cB_0 .$$
So
$$I=\frac12\epsilon_0c^3B_0^2 .$$
**Numbers.** $\epsilon_0=10^{-11}$ C²/(N m²), $c=3\times10^8$ m/s, $B_0=10^{-8}$ T:
$$I=\frac12\times10^{-11}\times(3\times10^8)^3\times(10^{-8})^2=\frac12\times10^{-11}\times2.7\times10^{25}\times10^{-16}=1.35\times10^{-2}\text{ W/m}^2 .$$
**Power through the circle of radius 1 m** (area $\pi$ m²):
$$P=I\times\pi=0.0135\,\pi\text{ W}\;\Rightarrow\;\frac{10^3P}{\pi}=10^3\times0.0135=\mathbf{13.5}.$$