An insulating sphere of radius a, centered at the origin, has a uniform volume charge density ρ.
A spherical cavity is excised from the inside of the sphere. The cavity has radius a/4 and is centered at position h(vector) , where |h(vector) |<(3/4)a, so that the entire cavity is contained within the larger sphere. Find the electric field inside the cavity.
Express your answer as a vector in terms of any or all of ρ (Greek letter rho), ϵ0, r(vector) , and h(vector) .
Here is the photo:
Electric Flux = ∫ E dot dA = Qencl / ϵ0
The Attempt at a Solution
The first part of the problem asked that we find the electric field inside the sphere with no cavity (for r<a) in terms of the position vector r. I calculated this by using Gauss' Law where dA = 4*pi*r^2 and q = ρ(4/3)r^3*pi. The correct answer was E = rρ/(3ϵ0)
I am running into trouble with this second part. If you set a Gaussian closed surface inside the cavity of the charged insulator, all of the electric field lines that go into it will not terminate inside it. Secondly, no electric field lines originate in the cavity. Therefore, the total charge inside the surface should be 0 and the electric field should be zero. This, however, is not the correct answer (It says it would be if the sphere were a conductor, but it is an insulator). I do not understand why this is so.