Finding the electric potential on a point on a sphere?

[miamirulz29]

Homework Statement

Consider a solid conducting sphere with an
inner radius R1 surrounded by a concentric
thick conducting spherical shell which has an
inner radius R2 and outer radius R3. There is
a charge Q on the sphere and no net charge
on the shell.
For all parts of this problem, we adopt
the standard convention of setting the electric
potential at infinity to zero.

Find the potential at Point A.
Find the Potential at Point O.

I have posted the problem and the choices. it is questions 10 and 11.

Homework Equations

V= kQ/r

The Attempt at a Solution

I think that the potential at point A is: 3. Va= KQ/a
For Point O: either 0 or infinity

Can somebody please tell me if I am correct. Thanks in advance.

 

[collinsmark]

For Point O: either 0 or infinity

Can somebody please tell me if I am correct. Thanks in advance.

No I don't think it is either 0 or infinity.

Remember, potential is related to the definite integral of the electric field.

I have a helpful hint. After almost forgetting it once and struggling miserably, I now repeat it myself almost every day. A definite integral is the area under the curve between two points.

I'm sure you already know that. But really think about it. It's so easy to forget what it really means.

In this problem, plot E as a function of r (make r the x-axis or something). You'll notice that E is curvy sometimes, and sometimes drops to zero and back. But how does that affect the area under the curve, from the point of infinity back to some point O?

Sometimes E drops to 0 for some region. How does that effect the potential? Well, "how does it affect the area under the curve from infinity to a point in that region?" is a better question. Since E is zero, the total area under the curve is constant, when integrated from infinity to a point in that region. But there is still area, even if the total area doesn't change! So the total area is not zero. And it's not infinity either.

 

[miamirulz29]

Sorry, this a non-calculus based physics course and I do not know calculus yet. However, I have figured out the answers. For 10: it is kQ/a and For 11: it is the one with three terms. I don't understand how the answer those two answers work. Could you explain them to me please? Thanks in advance.

 

[collinsmark]

Sorry, this a non-calculus based physics course and I do not know calculus yet. However, I have figured out the answers. For 10: it is kQ/a and For 11: it is the one with three terms. I don't understand how the answer those two answers work. Could you explain them to me please? Thanks in advance.

Ooh. No calculus it is then.

Okay, for problem 10, point a is on the outside of the whole thing. Whenever you end up on the outside of something with spherically symmetric charge distribution, simply treat the whole thing as a point charge. Just make sure the charge distribution is spherically symmetric, and the point of interest (the test charge) is on the outside. You can do this as a result of Guass' law. So that's where you get the kQ/a.

Problem 11 is a little trickier. The electric field inside a conductor is always zero. What's more, the electric potential inside a conductor is always constant. There are two conductors here to worry about. There is the solid sphere (where the test charge is at) and the spherical shell.

Let me start by commenting on the solid sphere by itself. You know that the potential anywhere inside the sphere is a constant. So simply measure the potential at the perimeter of the sphere, where you can treat the sphere as a point charge, and you know that the potential will be the same value anywhere inside.

If there was no conducting shell, this would be just like the previous problem. So as a first step, find the potential at R₁, ignoring the spherical shell for the moment.

For the next step we need to deal with the spherical shell. The potential is the same anywhere between R₂ and R₃ (the potential within a conductor is a constant). So we need to subtract its contribution. We know that the potential at point R₂ is the same as the potential at point R₃. If the shell was not there in the first place, what would this potential difference be between these two points? It would be
V(R₂) - V(R₃)
So calculate what that is. This is what we need to subtract before we finish the problem.

So now put everything together. Start with the potential of the solid sphere, as if there was no shell around it. Then from that, subtract the potential caused by inserting the conducting shell (i.e. subtract the result obtained in the above paragraph).

[Edit: Depending on how you approach this problem, you can either subtract [V(R₂) - V(R₃)] as discussed above, or add [V(R₃) - V(R₂)], which gives you the same answer. Both are valid methods. But subtracting the [V(R₂) - V(R₃)] might be more appropriate depending on how your text and instructor present this material. I went back and forth on editing this post, and decided on the subtracting method in the end, since I feel it is more in line with the calculus approach.]