What Determines the Electric Field Inside a Spherical Cavity?

[rockbreaker]

Hi, I am dealing with a problem in Electrostatics.

Homework Statement

There is a sphere with a spherical cavity in it. The sphere itself does not have net charge, but inside the cavity, there is a point charge at the center of the cavity. What's the electric field inside the cavity?

Homework Equations

Gauss' Law, E=Q/(4*Pi*epsilon0*r^2)

The Attempt at a Solution

I assume the field is 0 because the induced charges on the cavity surface cancels the field of the point charge in it. Is this assumption correct?

 

[NihalSh]

I assume the field is 0 because the induced charges on the cavity surface cancels the field of the point charge in it. Is this assumption correct?

No, the assumption isn't correct. The induced charge in fact support that there is an electric field inside the cavity. try to work it out

 

[collinsmark]

Hi, I am dealing with a problem in Electrostatics.

Homework Statement

There is a sphere with a spherical cavity in it. The sphere itself does not have net charge, but inside the cavity, there is a point charge at the center of the cavity. What's the electric field inside the cavity?

Homework Equations

Gauss' Law, E=Q/(4*Pi*epsilon0*r^2)

The Attempt at a Solution

I assume the field is 0 because the induced charges on the cavity surface cancels the field of the point charge in it. Is this assumption correct?

That assumption isn't correct for the electric field inside the cavity (inside of the surface containing the induced charges).

If it helps to understand why, remember what the electric field is inside of a uniformly charged, spherically symmetric shell without any other charges in it. Then consider what the electric field is if there is an additional charge at the center, all else being the same.

Now use Gauss' Law to apply that to this problem with the cavity. You shouldn't even have to assume that the cavity is symmetric -- it can be of any shape. Also, the cavity, and the charge inside of it, doesn't need to even be centered in the conducting sphere -- it can be anywhere within the sphere.

After you answer this step, it's the next steps that end up being rather fascinating and insightful. You will be invariably asked next to find the electric field within the conducting sphere, but outside the cavity, and then to find the electric field outside of the entire conducting sphere.

Hint: At first this might seem like a very mathematically intense exercise, but it is not. If you use Gauss' Law it turns out to be a surprisingly simple problem (well, comparatively simple for an electrostatics problem anyway). Most of this exercise is thinking about Gauss' Law qualitatively.

 

[rockbreaker]

You're right, I actually confused this problem with a dielectrical sphere. It is pretty obvious using Gauss' Law that there has to be an electric field inside the cavity when there is a charge inside it (which ends at the surface of the cavity to guarantee that the conductor is field-free). Thank you very much!

 

[Nickie]

Hello,

Thank you for sharing your problem in Electrostatics. Your assumption is correct, the electric field inside the cavity will be 0.

This can be explained by Gauss' Law, which states that the electric flux through a closed surface is proportional to the enclosed charge. In this case, the sphere does not have a net charge, so the electric flux through its surface is 0. This means that the electric field inside the cavity must also be 0, as there is no enclosed charge to produce an electric field.

Furthermore, as you mentioned, the induced charges on the cavity surface will cancel out the electric field of the point charge at the center of the cavity. This is due to the fact that the electric field lines from the point charge will terminate on the induced charges, resulting in a net electric field of 0 inside the cavity.

I hope this helps to clarify your understanding of the electric field in a cavity. Keep up the good work in Electrostatics!