Below is a question given in my assignment. I tried applying Gauss law in both forms, differential and surface integral form. But both there is a difference by a factor of $2$. Is the differential form not applicable here?
There is a spherical charge distribution where Potential varies as $V = V_0r^3$. We have to find $\rho(r)$, ie charge density.
$\mathbf{Try\ \ 1}:\ \ \ \ \nabla\cdot\vec{E} = \dfrac{\rho}{\epsilon_o}$
Since $V = V_o r^3$,
$$\vec{E} = -3V_or^2\hat{r}$$
$$\dfrac{\partial(-3V_o r^2)}{\partial r} = \dfrac{\rho}{\epsilon_o}$$
$$\boxed{\rho(r) = -6V_o \epsilon_o r}$$
$\mathbf{Try\ \ 2}:\ \ \ \ \Phi=\int_S\vec{E}\cdot d\vec{A}$
Since $V = V_o r^3$,
$$\vec{E} = -3V_or^2\hat{r}$$
$$\phi = \vec{E}\cdot 4\pi r^2 \\ =-12V_o\pi r^4$$
$$\implies Q_{enc}=-12V_o\epsilon_o\pi r^4$$
Now, $\rho = \dfrac{dQ}{dV}$, where V is Volume.
$$\implies \rho = \dfrac{-48V_o\epsilon_or^3dr}{4\pi r^2 dr}\\
\boxed{\rho(r)= -12V_o \epsilon_o r}$$
Q. Is the differential form not applicable here? or am I applying it incorrectly?
I do not seek answer to numerical question, only whether the approach is correct.
Best Answer
The differential and integral forms should in principle always lead to the same result, since they are related to each other via Gauss's theorem. (e.g. see What are the differences between the differential and integral forms of (e.g. Maxwell's) equations? ).
In this case you have not applied the differential form correctly, because you have used an incorrect form for divergence in spherical polar coordinates.