Average field inside spherical shell of charge

[chimay]

TL;DR Summary

Calculation of the electric field inside a spherical shell of charge with non-uniform density

A known result is that the average field inside a sphere due to all the charges inside the sphere itself is proportional to the dipole momentum of the charge distribution (see, for example, here).

I wonder whether the same result can be applied in the case of a spherical shell of non-uniform charge density. Is there any result about the average electric field inside the shell?

 

[Nugatory]

I wonder whether the same result can be applied in the case of a spherical shell of non-uniform charge density. Is there any result about the average electric field inside the shell?

It depends on whether the non-uniform density is still spherically symmetrical.

 

[anorlunda]

Can't you answer the question with two point charges at different locations plus superposition?

 

[chimay]

Thank you both of you for your replies. I understand it is a matter to average the contribution of each elemental charge on the sphere surface and then sum all of them, but I am not able to go through all the calculations. For that reason, I was thinking to apply the result that I mentioned in my first post to a sphere with radius slightly larger than that of the spherical shell, let's say $$ R_{sphere} = R_{shell} + \Delta R $$ with $\Delta R \rightarrow 0$. In this case I would get, again, that I can compute the average field inside the shell by computing the dipole momentum of the shell. Am I correct?

From a practical standpoint, I am working with a double polarity charge density. Let's say something like $\sigma = A \cos{\theta}$, to fix the ideas.PS: It seems I am not able to render the latex code in my reply. Can you render it correctly or did I make anything wrong?