Question

Consider a function $\overline{f}=\frac{1}{r^2}\hat{r}$, 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

• A. 0
• B. $2\pi$
• C. $4\pi$
• D. $R\pi$

Answer 1

The correct option is C $4\pi$

Ans : (c)

$\overline{f}=\frac{1}{r^2}.\hat{r}$
From divergence theorem as we know,
${\int }_{vol.}(▽.\overline{f})dV$=$∯$${}_s$$\overline{f}.d\overline{S}$
$∯$${}_s\overline{f}.d\overline{S}$=
$∯$${}_S(\frac{1}{r^2}.\hat{r})\times r^{2}sin\theta .d\theta .d\varphi .\hat{r}$
=$∯$${}_{S}sin\theta .d\theta .d\varphi =4\pi$