## Electric field in a sphere with a hole

As the title implies, I am having a problem with the way this problem is done.

problem: "An insulating sphere with radius a has a uniform charge density p. The sphere is not centered at the origin but at vector(r) = vector(b). Show that the electric field inside the sphere is given by E = p(vector(r) - vector(b))/3epsilon-naught.

An insulating sphere of radius R has a spherical hole of radius a located within its volume and centered a distance b from the center of the sphere, where a < b < R (a cross section of the sphere is shown in fig 22.42). The solid part of hte sphere has a uniform volume charge density p. Find hte magnitude and direction of the E field inside the hole, and show that E is uniform over the entire hole. ( Hint: use the principle of superposition and the result of part(a)).

Alright, I have part a figured out and understand how the two vectors can be used to find the radius of the sphere by subtraction. From this I assumed it was implied that r was a vector from the origin to the outside of the sphere while b was from the origin to the center. Subtraction yields the radius. As for fig 22.42 it is a circle with radius R, charge density p, and the hole to the right with what looks to be a center on the x axis. b is the distance to it from the center and a is its radius.

Now onto finding the electric field. I know that subtracting the E field of that region from the entire field will give me the rest of the E field with the use of Gauss's law. What troubles me is in two parts. First, how is the E field in the hole uniform? I would expect the lack of the volume of the sphere to cause one side of the sphere to provide a larger E field than the other. Also, wouldn't the E field inside the hole be different at different parts? I know my teacher mentioned if you take out a second hole than the E field is not uniform. Second, how would you even define one E vector for the sphere and one for the hole if it is a large sphere made up of many particles?

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 first, there is more volume on one side then on the other; however, the volume on the other side is closer. we know that there are an infinite about of vectors in this sphere, so how would you add them all up?