Electric Potential of grounded hollow metal sphere

[ti-84minus]

Homework Statement

There is a solid metal ball of radius 'a' (with charge Q) that is placed inside a hollow metal sphere (inner radius = 'b', outer radius = 'c'). The metal sphere is grounded (so V = 0 at r = c).
a) Find the electric potential V everywhere.
b) Find the charge on the metal shell at r = c and r = b.

Homework Equations

V = U/q₀.
V = $\frac{1}{4\pi\epsilon_{0}}\int\frac{dq}{r}$

The Attempt at a Solution

I considered the problem case by case:
r<a : V = $\frac{Q}{4\pi r\epsilon_{0}}$

a<r<b
b<r<c
r>c

Would the other three cases have the same result, since electric field is zero when b<r<c and r>c? (actually, when r>c, IS the electric field zero, since the sphere is grounded?)

Thanks for your help!

 

[anathema]

Thank you for your post. I would like to provide some input on your question.

First of all, your equations for electric potential are correct, but I would like to clarify the meaning of the symbols. V represents the electric potential, U represents the potential energy, q0 represents the test charge, and dq represents an infinitesimal charge element. Also, I would like to point out that the electric potential is a scalar quantity, not a vector as the electric field is.

Now, to answer your question, yes, the electric field is indeed zero inside the grounded metal sphere. This is because the electric field inside a conductor is always zero, and since the metal sphere is grounded, it is considered as a conductor. Therefore, all four cases in your solution will have the same result, which is V = Q/4πε0r.

For part b), the charge on the metal shell at r = c will be equal to the charge on the inner metal ball, which is Q. This is because the metal shell is grounded, so the charge on the inner metal ball will spread out evenly over the entire surface of the metal shell.

At r = b, the charge on the metal shell will be equal to the charge on the inner metal ball minus the charge on the hollow space between r = a and r = b. This can be calculated using the equation for electric potential that you have provided.

I hope this helps to clarify your understanding of the problem. If you have any further questions, please do not hesitate to ask.