Question

The electric potential in a region is given as $V=-4a{r}^2+3b$ , where $r$ is distance from the origin, $a$ and $b$ are constants. If the volume charge density in the region is given by $\rho =na{ϵ}_0$ , then what is the value of $n$ ?

Answer 1

Answer: 24

$V=-4a{r}^2+3b$
$E=\frac{-dV}{dr}=8ar$
The small amount of flux $d\varphi$ associated with the closed surface bounded by two spheres of radius $r$ and $r+dr$ is
$d\varphi =\left(E+dE\right)\left(A+dA\right)-EA$
$d\varphi =EdA+dEA$
$dE=8adr$ and $dA=8\pi rdr$
Using Gauss's law
$\frac{\rho \times 4\pi r^{2}dr}{{ϵ}_0}=8adr\times 4\pi r^2+8ar\times 8\pi rdr$
⇒ $\rho =24a{ϵ}_0$
$n=24$