Electric field in a sphere with a hole

[Sculptured]

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?

 

[6Stang7]

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?

 

[kyle8921]

I would like to address your concerns and provide some clarification on the problem at hand. First of all, it is important to understand that the electric field inside a sphere with a hole is not uniform in the sense that it has the same magnitude and direction at every point. Rather, it is uniform in the sense that it is constant throughout the entire hole.

To understand this, let's break down the problem into smaller parts. We know that the electric field inside a solid sphere with uniform charge density is given by E = p(r-b)/3ε0, as you have correctly stated. This means that at any point inside the solid sphere, the electric field will have the same magnitude and direction, determined by the distance from the center of the sphere.

Now, when we add a hole to the sphere, the electric field inside the hole will be affected by the presence of the hole and the surrounding charge distribution. However, if we use the principle of superposition, we can break down the problem into two parts: the electric field due to the solid sphere and the electric field due to the hole. Adding these two fields together will give us the total electric field inside the hole.

But why is the electric field inside the hole uniform? This is because the electric field due to the hole will have a direction and magnitude that cancel out the electric field due to the solid sphere at every point inside the hole. This is why the net electric field inside the hole will be constant throughout.

As for your second concern about defining one electric field vector for the entire sphere and one for the hole, it is important to remember that we are dealing with a continuous charge distribution. This means that we can consider the sphere as being made up of infinitely small charged particles, each contributing to the electric field at a specific point. By using calculus and integrating over the entire sphere, we can find the total electric field at any point inside the sphere, including the hole.

In summary, the electric field inside a sphere with a hole is not uniform in the traditional sense, but rather it is constant throughout the hole. This is due to the superposition of the electric fields from the solid sphere and the hole. And we can define one electric field vector for the entire sphere and one for the hole by considering the continuous charge distribution of the sphere. I hope this explanation helps to clarify the problem for you.