Question

Divergence of the three dimensional radial vector field $\overrightarrow{r}$ is

• A. $3$
• B. $1/r$
• C. $\hat{i}+\hat{j}+\hat{k}$
• D. $3(\hat{i}+\hat{j}+\hat{k})$

Answer 1

The correct option is A $3$

Let the radial vector is $\overrightarrow{r}$.
Divergence in spherical co-ordinate system
$▽.\overrightarrow{r}=\frac{1}{r^2}\frac{\partial }{\partial r}(r^2.r)=\frac{1}{r^2}.3r^2=3$

Alternatively:
$\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$
$▽.\overrightarrow{r}=\frac{\partial }{\partial x}\left(x\right)+\frac{\partial }{ry}\left(y\right)+\frac{\partial }{\partial z}\left(z\right)$
$=1+1+1=3$