GATE 2014 EC (EC1) – Question 63
In spherical coordinates, the far fields of a free-space dipole are $\mathbf E=(100/r)\sin\theta\cos(\omega t-\beta r)\hat a_\theta$ V/m and $\mathbf H=(0.265/r)\sin\theta\cos(\omega t-\beta r)\hat a_\phi$ A/m. Find average power (W) through the hemispherical shell $r=1$ km, $0\le\theta\le\pi/2$.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 55.4 to 55.6
Explanation
The average radial Poynting density is(1/2)(100/r)(0.265/r)sin²θ=13.25sin²θ/r² W/m². Multiply by spherical area r²sinθ dθdφ and integrate φ from0to2π and θ from0toπ/2. Since ∫sin³θdθ=2/3 over that interval, P=13.25×2π×2/3=**55.5015W**. Radius cancels.