Question about example 3.9 in griffiths EM

[bulgakov]

This is not actually a homework question, just something I am wondering about. A specified charge density dependent on /theta is glued over the surface of a spherical shell and you are asked to find the potential inside and outside the sphere, which is done in the example using separation of variables.

My question is - why can't you use Gauss's Law to find E and then find V from it (I see that wouldn't give the right answer, but why?)? Is it because the charge density depends on the angle? So Gauss's Law can only be used when the only dependence in the problem is on the radius?

 

[gabbagabbahey]

Hi bulgakov, welcome to PF!

My question is - why can't you use Gauss's Law to find E and then find V from it (I see that wouldn't give the right answer, but why?)? Is it because the charge density depends on the angle? So Gauss's Law can only be used when the only dependence in the problem is on the radius?

Well, in order to pull $\textbf{E}$ outside of the integral in Gauss' Law, there need to be certain symmetries present in the field. But the field takes on the symmetries of the charge density, so you need to have those same symmetries present in the charge density...is the charge density spherically symmetric in this case? Are there any other symmetries that would allow you to choose a Gaussian surface where $\int \textbf{E}\cdot d\textbf{a}=|\textbf{E}|A$? If not, then Gauss' Law isn't very useful in this case.

 

[bulgakov]

Thanks!