GATE 2022 CH – Question 45
Consider a sphere of radius 4, centered at the origin, with outward unit normal $\hat{n}$ on its surface $S$. The value of the surface integral $\iint_S\left(\frac{2x\hat{i} + 3y\hat{j} + 4z\hat{k}}{4\pi}\right) \cdot \hat{n}\,dA$ is ___________ (rounded off to one decimal place).
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Correct answer: 191.04 to 192.96
Explanation
By the divergence theorem the integral equals the volume integral of the divergence. The divergence is $\frac{2 + 3 + 4}{4\pi} = \frac{9}{4\pi}$, and the volume of the sphere is $\frac{4}{3}\pi \times 4^3 = \frac{256\pi}{3}$. The value is $\frac{9}{4\pi} \times \frac{256\pi}{3} = 192.0$.