Hollow Cavity within a charged sphere

[Lancen]

Suppose I had a uniformly charged sphere of some density, inside this sphere at some position there is a cavity of some radius. If I wanted to then find the Electric field somewhere, like say the surface of the sphere, how would I approach the problem?

Heres what I did. I pretended the cavity didn't exist and using Gauss' Law found the electric field at the surface of the sphere to be p*R/3*e, where p is charge density, R is the radius of the sphere and e is the electrical permittivity. Then I found the electric field for the cavity which has a radius of R/2 and is located in such a way that the distance from its center to the surface of the charged sphere is 3/2*R. and that was 4.5*(p)*R/e. Thats where I am stuck. As I was doing this I thought I could treat this like those moment of inertia problems where you have a piece of a wheel or something missing and you just treat that piece as negative mass. But then when I went to do this, I realized all I would be doing if I subtracted the field off off the cavity would be adding a negative field. Also, if it is just a cavity is there even a field inside it?

 

[Nate810]

Yes, there will be an electric field inside the cavity. The electric field inside a cavity in a uniformly charged sphere is just the same as the electric field outside the sphere, and can be calculated using Gauss' Law.

 

[m2003]

Your approach to finding the electric field at the surface of the charged sphere is correct. However, when considering the electric field within the cavity, you would need to take into account the electric field contributions from both the charged sphere and the cavity itself. This can be done using the principle of superposition, which states that the total electric field at a point is the vector sum of the individual electric fields from each source.

In this case, the electric field inside the cavity would be the vector sum of the electric field from the charged sphere and the electric field from the cavity itself. Since the cavity has no charge, its electric field would be zero. Therefore, the total electric field inside the cavity would be the same as the electric field from the charged sphere alone.

As for your comparison to moment of inertia problems, it is not accurate to treat the cavity as having negative charge. In this case, the cavity is simply a void within the charged sphere and does not have any charge itself. Therefore, subtracting the field from the cavity would not result in adding a negative field, but rather in removing the contribution from the cavity altogether.

In summary, to find the electric field at the surface of the sphere, you can use Gauss' Law as you did. To find the electric field inside the cavity, you would need to consider the electric field contributions from both the charged sphere and the cavity itself using the principle of superposition.