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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.

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    $\begingroup$ 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. $\endgroup$ May 10, 2019 at 2:47
  • $\begingroup$ 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. $\endgroup$
    – ACuriousMind
    May 10, 2019 at 16:15

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That's true, the net charge is zero, but there's no problem with that. It doesn't mean that the electric field is zero everywhere.

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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.

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