I Average field inside spherical shell of charge

AI Thread Summary
The discussion centers on the average electric field inside a spherical shell of non-uniform charge density and whether the known result for uniform charge distributions applies. It highlights that the outcome depends on the symmetry of the charge density. The concept of using superposition with point charges is suggested for analysis. The original poster proposes calculating the average field by considering a sphere slightly larger than the shell and using the dipole momentum approach. Practical examples of charge density, such as a double polarity density, are also mentioned, indicating the complexity of the calculations involved.
chimay
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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?
 
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chimay said:
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.
 
Can't you answer the question with two point charges at different locations plus superposition?
 
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?
 
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