[Physics] Charge density in concentric spheres

electrostaticshomework-and-exercises

Question:

If there are two conducting spherical shells and the inner shell is grounded, what will be >the charge density in the inner shell if there is a charge Q placed on the outer shell?

Yes, this is a HW problem, but I am not asking you guys to solve it for me… just show me the way 😀

If there is a charge on the outer shell with radius $a$, the charge density will be $\frac{Q}{4\pi a^2}$.

That should induce, a charge density on the inner side of outer shell — $\frac{Q}{4\pi (a-x)^2}$ where $x$ is thickness of the outer shell.

Now, if the inner shell was never grounded, $\frac{Q}{4\pi b^2}$ ($b$ = radius of the inner shell) charge density would have been induced on the inner shell, now that it is grounded there would be no charge on it, right?

There is no need for a charge to be induced for the electric field to be 0 inside the inner shell.

This seems to be my conclusion, but I fear it is too simple for the question. There might be something I am missing.

Edit: the hint of the problem says when a sphere is grounded, potential is infinity.

Best Answer

Now, if the inner shell was never grounded, $\frac{Q}{4\pi b^2}$ ($b$ = radius of the inner shell) charge density would have been induced on the inner shell

If the inner shell were not grounded, it wouldn't be connected to anything. And if it wasn't connected to anything, how would it be able to have any charge density other than zero? Where would the extra charge come from?

now that it is grounded there would be no charge on it, right?

No, as pointed out in the comments, the potential is zero, but that doesn't mean the charge is zero.

Can you figure out the potential of the outer sphere? Once you know the potentials of both spheres, what else can you calculate using that information?

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