# Electric field-Charge inside a metallic shell

1. May 19, 2013

### Vibhor

I am having few doubts while studying electric field which i am putting here

Suppose there is a spherical metallic shell with point charge q1 at the center and q2 charge outside the sphere in space ...

1) Now what I have have read in this forum and on other places is that the induced charges rearrange themselves in such a way that they cancel the external electric field such that the electric field within the conducting shell is zero .

Now ,when we say external electric field ...do we mean electric field due to q2 as well as q1 ?

2) The force experienced by charge q1 placed at the center of the shell will be zero .

I am not sure about the reason behind this .The force can be zero only when the electric field at the center of the shell is zero .

Why is electric field zero at the center of the shell .What about the effect of outside charge q2 on q1 ?

I would be highly grateful if somebody could help me in clearing the concepts .

Thanks

Last edited: May 19, 2013
2. May 19, 2013

### Simon Bridge

That would be q2 in this case - the statement is a general consequence of having mobile charges in the conductor. The effect of any electric field anywhere on a conductor is to move the charges around. At static-equilibrium the net field inside the conducting region is zero. In this case, that would be the field inside the physical metal of the shell.
A consequence if this is that the field inside the cavity is also zero - you should be able to see this by considering the charge distribution required for the field to be non-zero somewhere in the cavity.

Yes.
The answer to both questions is the same: the charge on the surface of the shell has rearranged to cancel it out.

The electric field due to q2, everywhere inside the shell, is zero.

But the electric field due to q1 inside the shell is not zero.

3. May 20, 2013

### Vibhor

Hello Simon

Thanks for responding.Please reconsider your statement that by external field we mean field due to only the outside charge q2 .Why wouldnt q1 not play any role in inducing charges within metal of the shell ?

What if q2 hadnt been there?In that case dont you feel that induced charges in the metal would rearrange themselves so as to cancel the effect of electric field produced by central charge q1.

The charge q1 will induce -q1 negative charge on the inside surface of the shell and hence +q1 charge on the outside surface of the shell .

Now the electric field is not zero within the shell .

Please give me the reason why electric field is zero at the center of the shell .One reason I got was due to symmetry of charge distribution on the inside surface of the shell.

Another reason I got was because the shell is an equipotential surface .

Which of these reasons is correct ? or is there some other concept involved .

Thanks

4. May 20, 2013

### WannabeNewton

The electric field in the material of the conductor vanishes. If you have a spherical metallic ball and carve out a large concentric spherical cavity within the ball so that you end up with a thin metallic shell with an inner and outer surface and a large concentric spherical hollow cavity below the inner surface, and place a point charge in the cavity, then you will end up with what you are talking about, sort of like this: http://s3.amazonaws.com/answer-board-image/74fe503d-8d9d-47b9-a467-9674312e7cdd.gif (replace the insulator with a point charge)

So the electric field is zero in the material of the conductor, which is the region in between the inner and outer surface of the metallic shell. It is not zero in the spherical cavity where the point charge resides because this is not part of the material of the conductor. If you assume $E = 0$ in the cavity then by Gauss's law
$\oint E\cdot dA = 0$ which implies $Q_{\text{enclosed}} = 0$ which is a contradiction because there is a point charge residing in the cavity.

What happens is that the presence of the point charge in the cavity causes the charges in the conductor to arrange themselves along the inner and outer surface of the associated metallic shell so as to make the field due to the point charge exactly cancel with the field due to the charges of the conductor within the material of the conductor i.e. in between the inner and outer surface. However, there is no implication that the field in the cavity must then vanish.

What is true is that the field due to the point charge outside of the conductor will not be able to penetrate the shell i.e. there will be no field due to the external point charge anywhere within the conductor nor in the cavity: the field will be killed off by the charges on the outer surface of the conductor. Similarly, the field due to the point charge in the cavity will also vanish for all points exterior to the cavity because it will be killed off by the charges on the inner surface of the conductor.

5. May 20, 2013

### Simon Bridge

You asked what was meant by the "external field" in the statement:
the induced charges rearrange themselves in such a way that they cancel the external electric field such that the electric field within the conducting shell is zero .

... charge q1 would produce a non-zero electric field inside the shell, but charge q2 would not.
Therefore the "external field" in the statement under discussion is that produced by q2 and not that produced by q1.

The field at the center of the shell in the setup you described is not defined as there is a charge q1 at the center of the shell. Please reread my original reply - you will find the words "due to" get used. I never said the field inside the shell you described would be zero.

It is the field due to q2 that is zero.
The field inside the cavity is that due to q1 alone.

The two reasons you have found are both correct.

Last edited: May 20, 2013
6. May 20, 2013

### Vibhor

Thanks WannabeNewton and Simon

If I rephrase my statement as the induced charges rearrange themselves in such a way that they cancel the electric field produced by q1 and q2 such that the electric field within the material of conducting shell is zero .

Am I right ?

7. May 20, 2013

### WannabeNewton

Yes, the charges in the material of the conductor will move around until the net electric field within the material of the conductor is zero.

8. May 20, 2013

### Simon Bridge

Yep - specify "conductor" though - the effect is not limited to conducting shells.

The original statement was regarding a special case that there was no charge enclosed in the shell.

9. May 20, 2013

### Vibhor

My first doubt is cleared.

Second thing I want to understand is how is force experienced by central charge q1 zero despite the presence of outside charge q2 ?

Is it because of spherically symmetrical charge distribution -q1 on the inner surface of the shell or because the shell is equipotential or both ? Please explain the correct reasoning .

Last edited: May 20, 2013
10. May 20, 2013

### WannabeNewton

For the point charge outside the conductor, note that the only way this external charge can exert a force on the point charge in the carved out cavity within the conductor is if the electric field lines of the external charge penetrate the material of the conductor so as to reach the charge in the cavity but if it penetrates the material of the conductor then there is a non-vanishing electric field in the material of the conductor which we know cannot be true in the electrostatic configuration.

Last edited: May 20, 2013
11. May 20, 2013

### Vibhor

Fine .This explains how external charge q2 doesnt affect the central charge q1.

But why and how is electric field at the center of the shell zero ,thus zero force on charge q1 ?

Is it because of spherically symmetrical charge distribution -q1 on the inner surface of the shell or because the shell is equipotential or both ?

Last edited: May 20, 2013
12. May 20, 2013

### WannabeNewton

The electric field at the center of the shell is not zero as Simon already pointed out! There is a non-vanishing electric field in the cavity that the point charge resides in but this doesn't mean the point charge feels a force because it is the one creating the electric field in the cavity.

13. May 20, 2013

### Vibhor

Fine . Things are getting cleared but more doubts are creeping in .

What about the fact that the field at any point in space is determined by the distribution of all the charges in the space .Right?

The central charge is being acted on by all charges in space, including the charge induced on the inner and outer part of the shell and the charge outside the shell.

What if the question asked is (assuming charge q2 is at some distance towards right of the shell )

The force on the central charge due to the shell is

1) towards left
2) towards right
3) upward
4) zero

My reasoning says it should be 4)zero but the correct answer is 2)towards right.

This question was asked in a test I appeared.

Last edited: May 20, 2013
14. May 20, 2013

### Simon Bridge

Sketch the situation and draw in the induced charges - what do you observe.

15. May 20, 2013

### Vibhor

-q1 charge is induced on the inner surface of the shell whereas +q1 on the outer surface of the shell .-q1 charge is uniformly distributed on the inner surface .Outer surface will not be uniformly distributed due to influence of charge q2 .

Is it correct?

16. May 20, 2013

### Simon Bridge

Kinda - did you sketch the situation?
You should have more detail - roughly where do the induced positive and negative charges on the outside of the shell lie in relation to the position of +q2?