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GATE 2015 EE – Question 14

Engineering Mathematics · Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem · 1 mark · Multiple choice

Consider a function $\vec{f} = \frac{1}{r^2}\hat{r}$, where $r$ is the distance from the origin and $\hat{r}$ is the unit vector in the radial direction. The divergence of this function over a sphere of radius $R$, which includes the origin, is

  1. 0
  2. $2\pi$
  3. $4\pi$
  4. $R\pi$

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Correct answer: (C) $4\pi$

Explanation

Away from the origin the divergence of $\frac{\hat{r}}{r^2}$ is zero, but at the origin it is a point source. The total over any sphere that includes the origin equals the flux through its surface, $\frac{1}{R^2} \times 4\pi R^2 = 4\pi$.