Two charged nonconducting sphere shells

In summary: This is given by ##\frac{Q(-Q)}{4\pi\epsilon_o}\left(\frac{1}{2a}-0\right) = -\frac{Q^2}{8\pi\epsilon_o a}##.In summary, the problem involves two nonconducting spherical shells of radius a, one with a charge Q and the other with a charge -Q, brought together until they touch. The electric field outside and inside the shells can be calculated using superposition and the potential satisfies Laplace's Equation. The work needed to move the shells far apart can be calculated either by considering the potential energy of the two shells or by calculating the force between two point charges. Both methods yield the same
  • #1
JSGandora
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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.
 
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  • #2
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?
 
  • #3
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?
 
  • #4
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.
 
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  • #5


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.
 

1. What are two charged nonconducting sphere shells?

Two charged nonconducting sphere shells refer to two hollow spherical shells made of insulating material that have been given an electric charge. The shells do not allow the charge to flow through them, making them nonconducting.

2. How are the charges distributed on the nonconducting sphere shells?

The charges on the nonconducting sphere shells are distributed evenly on the surface of the shells. This means that the charge is spread out uniformly across the entire surface of the shells.

3. What is the electric field between two charged nonconducting sphere shells?

The electric field between two charged nonconducting sphere shells is zero. This is because the electric field inside a hollow conducting sphere is zero, and the nonconducting shells do not allow the charge to flow through them to create an electric field.

4. How does the distance between the two charged nonconducting sphere shells affect the electric force between them?

The electric force between two charged nonconducting sphere shells is inversely proportional to the square of the distance between them. This means that as the distance increases, the electric force decreases. This relationship is known as Coulomb's Law.

5. Can the charges on the nonconducting sphere shells be changed?

Yes, the charges on the nonconducting sphere shells can be changed by using a charging method such as induction or conduction. This involves bringing a charged object in close proximity to the shells, causing the charges to redistribute and creating a new charge distribution on the shells.

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