Electric field inside the material of a hollow non-conducting sphere

[CsLevi]

I know that the E-field around a hollow non-conducting sphere charged with Q charge comes immediately from Gauss' Law but I'm wondering what the situation is if we somehow go inside the material, we make a very small hole through the material of the sphere and go inside it. What would there be the E-field?

 

[Meir Achuz]

If the charge is a uniform surface charge, then the field inside would be zero by Gauss's law.

 

[CsLevi]

Even inside the hole when being half outside, half inside the sphere?

 

[Meir Achuz]

Inside the hole, at radius R, the E field is one half Q/R^2.
The E field goes from Q^2/R to zero as r decreases through the hole.
The decrease in E takes place in a distance of the order of \Delta r=d, where d is the diameter of the hole.

 

[CsLevi]

So right on the surface it would be E = Q/ (4R^2 pi * epsilon_0), but when going through the hole, right inside the sphere it is already 0. Does it follow from Gauss' Law to be more obvious? E.g.: we choose the Gaussian sphere with radius R such that it goes through the charge layer. Thus, the total amount of charge inside the Gaussian sphere is Q/2 so E inside the hole is indeed 1/2 * Q/ (4R^2 pi * epsilon_0).

 

[Meir Achuz]

That happens anywhere at the surface, not just at the hole.

 

[CsLevi]

Interesting, thanks!