Question

The value of ∮r/r^3.da on a spherical surfacer r = R is 

(a) 4ΠR²  

(b) Zero

(c) 4Π

(d) infinity

Answer 1

One needs to evaluate surface integral $\oint_A \frac{\bold r}{r^3} d\bold S = \oint_A \frac{\bold r}{r^3} \bold n d S$, where A is the surface of the sphere of radius $R$.

Let us use spherical coordinates $\theta, \varphi$. The normal vector to the surface of the sphere at point $\bold r$ is simply $\frac{\bold r}{r}$ , and the surface element is $d S = R^2 \sin \theta d\theta d\varphi$.

$\oint_A \frac{\bold r}{r^3} \bold n d S= \int_0^{2\pi}d\varphi\int_0^\pi \frac{r^2}{r^4}|_{r=R} \cdot R^2 \sin \theta d\theta = 2\pi \cdot (-\cos \theta)|_0^{\pi} = 4 \pi$.

Hence, the answer is c)