Concept of electric field and hollow conductors

[Kaguro]

Homework Statement

An arbitrarily shaped conductor encloses a charge q and is surrounded by a conducting hollow
sphere as shown in the figure. Four different regions of space, 1, 2, 3, and 4, are indicated in the
figure. Which one of the following statements is correct?

(A) The electric field lines in region 2 are not affected by the position of the charge q.
(B) The surface charge density on the inner wall of the hollow sphere is uniform.
(C) The surface charge density on the outer surface of the sphere is always uniform irrespective of
the position of charge q in region 1.
(D) The electric field in region 2 has a radial symmetry.

Relevant Equations

None.

I think:

Due to charge q, there will be a field in region 1, very much dependent on position of q. The inner surface charge density of irregular conductor is also dependent on the position( so that it could cancel the field of charge and E=0 inside body of irregular conductor). The outer surface charge density on the other hand, is non-uniform, but independent of position of q. It just has to equal to +q anyhow.

The electric field in region 2, is then dependent on the shape of the irregular conductor but still independent of position of q. This will lead to non-uniform charge density on inner surface of shell conductor. All of this to make E=0 in region 3. This will induce charge q on outer surface(uniform and independent of position of q)

So:
A is true
B is false
C is true
D is false

But only one option can be true, and it is given that only C is true.

What's wrong?

 

[Delta2]

Me on the other hand I think all options are false. I really can't understand why C is true. Here is my argument why A and D are false which I think is correct. (I can't understand what is happening about B and C but at first glance they seem false too)

Because the arbitrary shape conductor is arbitratry it has a irregular surface charge density. This surface charge density depends on the position of the charge q. According to the integral form of Gauss's law the electric field in region 2 depends only on the charge densities on the interior of region 2 and it doesn't matter what happens in the outer sphere. On the interior of region 2 there are charge densities(the irregular charge density) that are not symmetric, hence D is false. Also A is false because the charge densities inside region 2 depend on the position of charge q.

 

[haruspex]

A is false because the charge densities inside region 2 depend on the position of charge q.

There are no charges in region 2, so I'm not sure what you meant by that.
The field lines in 2 depend on both the position of q and the induced distribution of charges on the irregular conductor. Is it not possible that these effects would cancel to produce a constant set of field lines in 2 as q moves?

See bullet 3 at https://en.m.wikipedia.org/wiki/Faraday's_ice_pail_experiment#Description_of_experiment.

 

[Delta2]

There are no charges in region 2, so I'm not sure what you meant by that.
The field lines in 2 depend on both the position of q and the induced distribution of charges on the irregular conductor. Is it not possible that these effects would cancel to produce a constant set of field lines in 2 as q moves?

See bullet 3 at https://en.m.wikipedia.org/wiki/Faraday's_ice_pail_experiment#Description_of_experiment.

Yes sorry I meant the charges in region 1.

What cancels out is the charge density in the inner surface of the irregular conductor with the charge density in its outer surface, and what's left is the charge q whose position should affect the field line in region 2. So I still think A is false.

 

[haruspex]

What cancels out is the charge density in the inner surface of the irregular conductor with the charge density in its outer surface,

You mean, separately within each patch of the shell? It cannot do that. The field generated by those two surface charges would then be normal to the surface everywhere within the material of the shell, so would not in general act to neutralise the field from the point charge.

 

[Delta2]

You mean, separately within each patch of the shell? It cannot do that. The field generated by those two surface charges would then be normal to the surface everywhere within the material of the shell, so would not in general act to neutralise the field from the point charge.

Well I just don't know then, all I had in mind is that by trying to apply Gauss's law (in integral form) in region 2, three charges are enclosed: the point charge q, the charge -q in the inner surface and the charge +q in the outer surface. The total charge enclosed is q+q-q=q and it "seems to be" irregularly distributed, that's why I think the E-field in this region depends on the position of the charge q.

 

[Delta2]

Btw by looking at the figures of that link of post #3, it seems indeed that in a sphere, though in the inner surface the distribution is non uniform, in the outer surface the distribution is uniform. Any clue why is that?

 

[haruspex]

Btw by looking at the figures of that link of post #3, it seems indeed that in a sphere, though in the inner surface the distribution is non uniform, in the outer surface the distribution is uniform. Any clue why is that?

Because the charge on the inner surface distributes itself just so as to neutralise the contained charge.

I assume you read this at the link I posted:
"the charge on the outside of the container is not affected by where the charged object is inside the container."

Consider two scenarios:

1. As in the question, but the conductor is earthed.
There will be no field outside, nor within the wall of the conductor. A charge -q on the inner surface neutralises the contained charge. There is no charge on the outer surface. If the contained charge is moved around, the charge on the inner surface redistributes to maintain neutralisation.

2. The conductor is not earthed and carries a charge q. There is no contained charge.
All the charge is on the outside of the conductor and adopts a distribution that depends on the shape of the outer surface.

Now add these two charge distributions together for the problem as given.

Ok, I do not know how to prove any of this, but it seems feasible.

 

[Delta2]

Because the charge on the inner surface distributes itself just so as to neutralise the contained charge.

I assume you read this at the link I posted:
"the charge on the outside of the container is not affected by where the charged object is inside the container."

Consider two scenarios:

1. As in the question, but the conductor is earthed.
There will be no field outside, nor within the wall of the conductor. A charge -q on the inner surface neutralises the contained charge. There is no charge on the outer surface. If the contained charge is moved around, the charge on the inner surface redistributes to maintain neutralisation.

2. The conductor is not earthed and carries a charge q. There is no contained charge.
All the charge is on the outside of the conductor and adopts a distribution that depends on the shape of the outer surface.

Now add these two charge distributions together for the problem as given.

Ok, I do not know how to prove any of this, but it seems feasible.

There are a couple of things I don't understand:

• How the inner charge density can neutralize by itself the effect of the contained charged. I thought it needs to cooperate with the outer charge density in order to achieve that
• How exactly earthing works. Why earthing removes the charge only from the outer surface and not from the inner surface too?

 

[haruspex]

There are a couple of things I don't understand:

• How the inner charge density can neutralize by itself the effect of the contained charged. I thought it needs to cooperate with the outer charge density in order to achieve that
• How exactly earthing works. Why earthing removes the charge only from the outer surface and not from the inner surface too?

It's the same answer to both.
A single shell of charge, suitably arranged, can neutralise any contained charge distribution. Further down the Wikipedia article, it explains this using field lines:
https://en.m.wikipedia.org/wiki/Faraday's_ice_pail_experiment#Explanation_using_electric_field_lines
Since no (net) field lines exist in the wall of the container, all field lines from the contained charge terminate at the inner surface.

Given the above, any other charge on the container will distribute on the outer surface in exactly the same manner as if the contained charge and the inner surface induced charge did not exist.
I.e., if there is a net charge Q on the container and a contained charge q, the charge on the inner surface is -q and on the outer surface Q+q.
If we let the contained charge discharge onto the container, there will now be no charge on the inner surface, but the outer surface charge will be unchanged, both in magnitude and distribution.

 

[Delta2]

Ok I think now I am covered completely on why B is false and C is true.

Using the fruits of the discussion between posts #3 and #10 we can also conclude that D is false: In region 2 the effect of the point charge q is completely neutralized by the inner surface charge density of the irregular conductor, leaving the outer surface charge density to "command" region 2. But this outer charge density is irregular hence D is false.
"A" might be true though because the outer charge density is independent of q and the inner charge density. It depends only on the shape of the irregular conductor right. So is "A" true afterall?

 

[haruspex]

Ok I think now I am covered completely on why B is false and C is true.

Using the fruits of the discussion between posts #3 and #10 we can also conclude that D is false: In region 2 the effect of the point charge q is completely neutralized by the inner surface charge density of the irregular conductor, leaving the outer surface charge density to "command" region 2. But this outer charge density is irregular hence D is false.
"A" might be true though because the outer charge density is independent of q and the inner charge density. It depends only on the shape of the irregular conductor right. So is "A" true afterall?

I agree with @Kaguro that A and C are both true.

 

[SammyS]

@Kaguro ,
I agree that your analysis of the situation is correct.

 

[Kaguro]

Thank you everyone for the valuable discussion. I really like reading deep insights.