Electric Field Inside a Hollow Sphere

[Heisenberg7]

Let's assume that we have a hollow sphere with holes at opposite ends of the diameter. What would be the field inside the hollow sphere? I know that we can look at this as the superposition of the hollow sphere without holes and 2 patches with opposite surface charge density. For some reason, in my book, they say that the electric field inside is zero. No explanation. I mean sure, the electric field from the hollow sphere without holes is zero, but what about the patches? Wouldn't they create a field inside the hollow sphere?

 

[TSny]

We need more information. Is the sphere a conductor? Does the sphere carry any charge? If so, how is the charge distributed? Are there any charges external to the sphere? How big are the holes? What is the specific example that is being discussed in your book? What book are you using?

 

[Heisenberg7]

We need more information. Is the sphere a conductor? Does the sphere carry any charge? If so, how is the charge distributed? Are there any charges external to the sphere? How big are the holes? What is the specific example that is being discussed in your book? What book are you using?

It's an insulator. Assume that charge is uniformly distributed across its surface and that it has a surface charge density $+\sigma$. There is an external charge that's travelling towards one hole. Assume that they are small (also, what happens as the radius of the holes increases?). The specific example that is being discussed in my book is this one:

Inside a hollow, nonconducting, uniformly charged sphere of mass $m_1$ and radius $r$, at the ends of one of the diameters we carve out 2 small holes. The charge of the sphere is $q_1$. At the start of the composition, the sphere is at rest and far away from it a charge of mass $m_2$ and charge $q_2$ starts travelling towards it with speed $v_0$. Charges $q_1$ and $q_2$ are of the same sign and the charge is travelling towards one of the holes and we want it to exit on the other side. How much time will the charge spend inside the sphere? Assume that there is no friction.

The book I'm using was written in another language so it wouldn't be of much help to you if I mentioned which one it is.

 

[TSny]

Thank you for clarifying the setup. I feel pretty sure that by calling the two holes "small", it is intended that you may take the electric field inside the sphere (due to the surface charge) to be essentially the same as if there were no holes.

 

[Heisenberg7]

Thank you for clarifying the setup. I feel pretty sure that by calling the two holes "small", it is intended that you may take the electric field inside the sphere (due to the surface charge) to be essentially the same as if there were no holes.

Oh, great. So, if the holes were larger, the effect would be larger? Could this be illustrated by for example slowly increasing the radius of the holes to the point at which the sphere slowly becomes a 1 dimensional ring? It just seems more intuitive that way.

 

[TSny]

Oh, great. So, if the holes were larger, the effect would be larger? Could this be illustrated by for example slowly increasing the radius of the holes to the point at which the sphere slowly becomes a 1 dimensional ring? It just seems more intuitive that way.

Yes. For non-negligible-sized holes, there will be some electric field inside the sphere that depends on the size of the holes. Also, the field outside the sphere is modified by the holes.

In this problem, the holes are assumed to be small enough that you can neglect these complications. The small holes exist only to allow the charge $q$ to pass through the interior of the hollow sphere.