# Potential in case of concentric shells

1. Dec 25, 2015

### gracy

1. The problem statement, all variables and given/known data
A,B and C are three concentric metal shells of radii a,2a and 3a .Shell A is the innermost and shell C is the outermost .Shell A is given a charge q and C is earthed.Find the final potential of shell B.

2. Relevant equations
$V$=$\frac{Kq}{r}$

3. The attempt at a solution
Charge q' is supplied from earth to shell.

Then charges will be induced

Now we have to find q'
Potential of shell C should be zero.
$\frac{K(q'+q)}{3a}$ - $\frac{Kq}{3a}$ +$\frac{Kq}{a}$ - $\frac{Kq}{a}$ +$\frac{Kq}{2a}$ =0

$\frac{K(q'+q)}{3a}$ - $\frac{Kq}{3a}$ +$\frac{Kq}{2a}$ =0

$\frac{K(q'+q)-Kq+\frac{3Kq}{2}}{3a}$=0

$K(q'+q)$$-Kq$+$\frac{3Kq}{2}$=0

$\frac{2K(q'+q)-2Kq+3Kq}{2}$=0

$2K(q'+q)$-$2Kq$+$3Kq$=0

$2K(q'+q)$+$Kq$=0

$2K(q'+q)$=$-Kq$

$2(q'+q)$=$-q$

$q'+q$=$\frac{-q}{2}$

$q'$=$\frac{-q}{2}$-q

$q'$=$\frac{-3q}{2}$

Now by putting value of q' we can find potential of shell B by all the charges present.Am I right till here?

2. Dec 25, 2015

### cnh1995

As per my understanding, you should consider "net charge" on the spheres. Potential on any sphere will be due to the net charge q on sphere A and q' on sphere C. In your first equation, your first term represents the potential of sphere C due to its own (net)charge, which is q'. Why have you taken q+q' then? Also, sphere B will have no net charge, hence will not contribute to any potential. So,why is there kq/2a in the first equation?

3. Dec 25, 2015

### theodoros.mihos

Shells are thin so there is no radial charge distribution. There is only surface charge distribution that have the same value by spherical symmetry. Electric field between a and c can calculate be Gauss law and we know that potential on c is zero.

4. Dec 25, 2015

### cnh1995

Sphere B is outside the sphere A but inside the sphere C. This means potential of B due to sphere C will be same as the potential of C due to its own (net)charge q'.

5. Dec 25, 2015

### gracy

Potential can also be due to induced charges.

6. Dec 25, 2015

### gracy

qi is supplied by earth and +q is charge induced due to -q present on inner surface of shell C.

7. Dec 25, 2015

### cnh1995

When charges are induced, both +ve and -ve charges appear on the sphere. When it is said that induced +ve charge flows to the ground, that means ground has supplied extra -ve charge to the body.

8. Dec 25, 2015

### cnh1995

+q and -q will form net 0 charge. It will be only q' supplied by the ground that will be responsible for potential.

9. Dec 25, 2015

### haruspex

Your expression for the potential at C is all wrong.
Hang onto these facts for uniformly charged conducting spherical shells:
- outside the shell, the potential and field generated by the charge on the shell is exactly the same as if the whole charge were concentrated at the centre of the sphere. (So what is the potential at C that results from the charge on A?)
- there is no field between the inner and outer radii of the shell, so the potential is constant in the annulus.
- there is no field generated by the charge on the shell inside the shell, so the potential it generates there is also constant and equal to the potential it generates at the shell itself. (Don't confuse this with the net potential at the shell.)

10. Dec 26, 2015

### Staff: Mentor

Gracy, if shell C were not grounded, do you agree that the charge induced on its outer surface by charge separation due to the charge on shell A would have been +q?

If so, then adding the ground connection would allow that +q charge to "escape" to ground. Or, thought of in another way, ground would supply a charge of -q to cancel that +q charge. The end result is that a charge of -q moves onto shell C through the ground path. This charge of -q serves to terminate the field lines of the interior charge +q of A shell, and so shell C exhibits a zero potential to the external world.

So you end up with shell A holding a net charge of +q, shell B holding a net charge of 0, and shell C holding a net charge of -q.

Given these shell charges you can apply the facts listed by @haruspex to find the potential at the location of shell B. Note in particular that an isolated uncharged shell (such as shell B) will not contribute to the potential. Only net charge on a shell will do so. The fields created by charge separation serve to cancel the field inside the conductor comprising the shell (so no currents continue to flow) and matches the external field at the boundaries of the conductor. That is to say, superposition of the external field and the field produced by the separated charges renders the net field inside the conductor (within the shell material) zero, but doesn't change the external field or potential.

In the end, the potential at the position of shell B will be the superposition of the potentials due to the net charges of shell A and C.

11. Dec 27, 2015

### gracy

How?I mean electric field lines of interior charge +q of A shell have to be terminated on inner surface of shell B as electric field can not exist in metal body hence electric field lines of charge +q of A shell would not able to go beyond inner surface of shell B.

12. Dec 27, 2015

### cnh1995

Electric field will not exist in the thickness(however small it is) of the sphere. So, when -q charge is induced on the inner surface of B, +q charge is induced on the outer surface. This makes the field 0 inside the thickness of the sphere. However, as now +q is on the outer surface of B, its field lines will reach out to sphere C. Here, there is a 'discontinuity' in the field due to the thickness of the sphere. The field wouldn't be trapped inside any sphere, if sphere C weren't grounded.

13. Dec 27, 2015

### haruspex

Yes, you can think of it as lines from A terminating on inner surface of B, but then being regenerated from the outer surface of B. Or, as gneill suggests, you can note that B has no net charge, so (as far anything except B itself is concerned) it is the same as the lines from A going all the way out to C. Work with whichever you are comfortable with.

14. Dec 27, 2015

### gracy

Then why there is potential due to induced charges in the video I have mentioned?

15. Dec 27, 2015

### cnh1995

The charge in the video is external to the sphere. Hence, to make the potential of every point inside the sphere same, there will be potential due to the induced charges. The net potential of every point inside the sphere will be kq/r. Out of that, kq/x will be due to the external +q and the rest will be due to the induced charges. In fact, to make the potential equal to kq/r, the charges will be induced.
In case of concentric spheres(your original question), the charge is inside the spheres B and C and 'on' the sphere A.
Consider only spheres A and B:
Charges will be induced on inner and outer surface of B to make the potential inside B equal at every point, thus making
Einside B=0.
So, if you pick any point inside sphere B at a distance x from the center, at that point, the potential will be kq/2a. Out of that potential, kq/x will be contributed by +q on sphere A and the rest will be contributed by the induced charges on sphere B. But
the net potential of sphere B is kq/2a. That is what is asked in your original question.
I haven't considered sphere C here. You can extend this logic to sphere C and get the final answer.

16. Dec 27, 2015

### cnh1995

In short, potential of the sphere is decided by the net charge present in the system(and radius of the sphere(s)). Induced charges only help to establish this pre-decided potential, wherever necessary. At the center of the sphere(in the video above), contribution due to the induced charges is 0 since only +q charge outside is sufficient( or +q is the one to decide the potential at the center) and the induced charges are symmetrical and equidistant from the center. Anywhere else inside the sphere, potential due to the +q outside is not kq/r. So, to make it kq/r, charges are induced on the sphere. So, in your original question, it is only the net charge that will decide the potential of the spheres. So, net charge is the one who will decide the potential and induced charges will simply help to execute the task of establishing that potential everywhere inside the sphere and make Einside=0.

Last edited: Dec 27, 2015
17. Dec 27, 2015

### gracy

I am not getting then how potential only depends on net charge present and not on induced charges .Isn't induced charge playing role equal to that of net charge present in determining potential?

Last edited: Dec 27, 2015
18. Dec 27, 2015

### cnh1995

Magnitude of the net potential is decided by the net charge. Induced charges appear to establish that potential inside the sphere, wherever necessary. So, role of induced charges is vital in "establishing" the net potential(which is already determined by the net charge) and not in determining it. Of course, the potential due to induced charges will vary at each point but the total potential at any point will remain same i.e. kq/r.

Last edited: Dec 27, 2015
19. Dec 27, 2015

### haruspex

It's hard to find words that distinguish between the cavity inside the inner surface of a shell and the region between the surfaces of the shell. I understand that here you are referring to the latter, but it might not be clear. I use the term annulus to describe this.

You could further elaborate that the potential in the annulus due to induced charges on the inner surface exactly cancel the potential due to A, while those on the outer surface produce the constant potential kq/2a.

20. Dec 27, 2015

### SammyS

Staff Emeritus