# Spherical surface charge distribution [closed]

I have a problem that sounds like this:

Consider a spherical surface of radius $$b$$ with the fixed charge distribution $$\sigma(\theta)=\sigma_0 \cos(\theta)$$, $$\theta$$ being the polar angle. Calculate the electric field inside and outside of the surface $$E_i(r,\theta), E_e(r,\theta)$$.

I know that, to find the charge, I have to integrate the charge distribution over the whole sphere. The problem is, when I integrate the distribution $$\sigma(\theta)=\sigma_0 \cos(\theta)$$ varying the $$\theta$$ from 0 to $$\pi$$ and the angle $$\phi$$ from 0 to $$2\pi$$, it is equal to zero.

• You are not asked to find the charge. You know the charge. You're asked to find the field. And yes the net charge is zero. May 10, 2019 at 2:47
• Please note that homework-like questions and check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual questions over those just asking for a specific computation. May 10, 2019 at 16:15

Obviously you are supposed to use the spherical multipole expansion. Notice that $$\sigma \propto Y^0_1$$. You now should be able simply to plug everything into the formulas to get the result.