Question

The electric field potential at a point in the space region depends only on x coordinates as $V=-a{x}^3+b$. The space charge density $\rho$(x) is $na{\epsilon }_{0}x$. Find $n$.

Answer 1

$V=-a{x}^3+b$
$\frac{dV}{dx}=-3a{x}^2$
$E_x=-\frac{dV}{dx}=3a{x}^2$
Using differential form of Gauss's law in one dimension, $\frac{\partial E_x}{\partial x}=\frac{\rho }{{ϵ}_0}$
or $\frac{\partial \left(3a{x}^2\right)}{\partial x}=\frac{\rho }{{ϵ}_0}$
or $\rho =6ax{ϵ}_0$