Two charged nonconducting sphere shells

[JSGandora]

Homework Statement

One of two nonconducting spherical shells of radius a carries a charge Q uniformly distributed over its surface, the other a charge -Q, also uniformly distributed. The spheres are brought together until they touch. What does the electric field look like, both outside and inside the shells? How much work is needed to move them far apart?

Homework Equations

$U = \epsilon_0 \int E_1 \cdot E_2 dv$

The Attempt at a Solution

I tried using $U = \epsilon_0 \int E_1 \cdot E_2 dv$ to bash out the work it takes to move the two spheres apart to infinity but it is way too messy.

 

[TSny]

Consider the electric field (inside and outside) of a single uniformly charged shell. For two shells, think superposition. Can you relate the problem to 2 point charges?

 

[JSGandora]

So the electric field by one of the spheres, say the shell of charge Q, is the same field as caused by a point charge Q at the center of that shell. So can we say that the potential outside the shell satisfies Laplace's Equation? Then the average value of the potential over the shell of charge -Q is the same as the potential at the center of the shell, which is $\frac{1}{4\pi\epsilon_o}\frac{Q}{2a}$, so the energy needed to move these two shells infinitely apart would be $\frac{Q^2}{8\pi\epsilon_o a}$. Is that correct?

 

[TSny]

Yes, I believe that's right.

Another way to approach the problem is to consider the force between the two shells. That should be the same as the force between two point charges with charges Q and -Q. So, the work that you want to calculate is just the change in potential energy of the two point charges as they are separated from a distance of 2a to infinity.

 

[HalfLostAndSearching]

A better approach would be to use the concept of electric potential energy. Initially, when the spheres are brought together, they have a potential energy of 0 since they are not interacting with each other. However, as they are moved apart, they start to experience a repulsive force due to their opposite charges. This repulsive force does work on the spheres, increasing their potential energy.

The potential energy of a point charge q at a distance r from another point charge Q is given by:

$U = \frac{1}{4\pi\epsilon_0} \frac{qQ}{r}$

For the first sphere, the potential energy due to the charge Q on the second sphere is given by:

$U_1 = \frac{1}{4\pi\epsilon_0} \frac{Q(-Q)}{2a} = -\frac{Q^2}{8\pi\epsilon_0a}$

Similarly, for the second sphere, the potential energy due to the charge -Q on the first sphere is also given by:

$U_2 = -\frac{Q^2}{8\pi\epsilon_0a}$

The total potential energy of the system is the sum of these two energies:

$U_{total} = U_1 + U_2 = -\frac{Q^2}{4\pi\epsilon_0a}$

As the spheres are moved apart, the potential energy of the system increases. When they are moved to infinity, the potential energy becomes 0 again, since they are no longer interacting with each other. The work done in moving the spheres to infinity is equal to the increase in potential energy, which is given by:

$W = U_{total} = -\frac{Q^2}{4\pi\epsilon_0a}$

Therefore, to move the spheres far apart, a work of $\frac{Q^2}{4\pi\epsilon_0a}$ is needed.

As for the electric field, outside the spheres, the electric field is the same as that of a point charge located at the center of the spheres, with a magnitude of:

$E = \frac{Q}{4\pi\epsilon_0r^2}$

Inside the spheres, the electric field is 0, since the charges on the inner surface cancel out each other's electric fields.