Volume density vs Surface density of charge distribution

[vcsharp2003]

Homework Statement

I have come across questions in Electrostatics where a solid sphere has a volume density of charge distribution mentioned, but in some questions an area density of charge is given for a solid sphere.

My question is, when will a solid sphere have a volume or area distribution of charge?

Relevant Equations

None

This doubt is confusing to me.

I know it's something to do with conductors and insulators, but cannot explain. Conductors have mobile/free electrons unlike insulators. Having free electrons doesn't seem to explain this difference of charge distributions.

 

[haruspex]

If it’s a conductor it will only have charge on the surface.

 

[vcsharp2003]

If it’s a conductor it will only have charge on the surface.

What would be the reason for this? Also, which material will have a volume density of charge?

 

[haruspex]

What would be the reason for this? Also, which material will have a volume density of charge?

Try page 48 of http://web.mit.edu/sahughes/www/8.022/lec05.pdf

 

[vcsharp2003]

Try page 48 of http://web.mit.edu/sahughes/www/8.022/lec05.pdf

That link provides excellent reading on electrostatics. Thank you.

The para on page 48 that seems to explain is as below. Is this correct?

"Net charge can only reside on the surface of a conductor. This is easily proved with Gauss’s law: make a little Gaussian surface that is totally contained inside the conductor. Since there is no $\vec E$ -field inside the conductor, $\oint \vec E \cdot d \vec A$ is clearly zero for your surface. Since that is equal to the charge the surface contains, there can be no charge."

 

[vcsharp2003]

Try page 48 of http://web.mit.edu/sahughes/www/8.022/lec05.pdf

Also, electric field inside a charged conductor is zero because if it was not so then mobile electrons in sphere would still move and redistribute due to an electrical force on them. This seems a good explanation for the case when a charged rod is brought close to solid spherical conductor so external field due to charged rod gets canceled by internal field created by charges on opposite ends of the sphere.

But the above explanation fails in following scenario. A positively charged rod is brought near a solid spherical conductor. After charges have redistributed on the surface of the sphere, the positively charged side opposite to the side next to charged rod is earthed. Earthing is removed now. Then the charged rod is taken away from the sphere as a last step so it has none of its electric field acting within the sphere. As a result, negative charges will redistribute on the surface of the sphere. Why will they stop redistributing after a moment in this type of scenario?

 

[haruspex]

Also electric field inside a charged conductor is zero because if it were not so then mobile electrons would still move and redistribute due to an electric force on them.

Quite so.

 

[vcsharp2003]

Quite so.

I have edited my post to which you replied above. I cannot explain using the same explanation why after earthing one side of the charged sphere and removing the charged rod from its vicinity, charges stop redistributing?

 

[vcsharp2003]

I have edited my post to which you replied above. I cannot explain using the same explanation why after earthing one side of the charged sphere and removing the charged rod from its vicinity, charges stop redistributing?

Perhaps, the electric field created by redistributed negative charges results in net zero force on electrons that are still inside the solid sphere since the electrons on surface are pushing the inside electrons in all directions and these forces cancel each other.

 

[haruspex]

the positively charged side opposite to the side next to charged rod

Makes no difference where the Earth is attached.

The charged rod is then taken away from the sphere

Is it still earthed?

Why will they stop redistributing after a moment

Once you stop changing the setup, charge redistribution will swiftly settle into equilibrium. Maybe I do not understand the question.

 

[vcsharp2003]

Is it still earthed?

The earthing is removed while the charged rod is held close to the other end of sphere. After this, the charged rod is removed from the vicinity of sphere.

 

[vcsharp2003]

Maybe I do not understand the question.

I was trying to use the explanation of external field cancelling the internal field to reason why electric field inside the conductor is zero. After earthing and charged rod are removed, how can we say net field inside the charged spherical conductor is zero.

 

[haruspex]

I was trying to use the explanation of external field cancelling the internal field to reason why electric field inside the conductor is zero. After earthing and charged rod are removed, how can we say net field inside the charged spherical conductor is zero.

Because when the rod is removed the (negative) charges on the sphere will redistribute to be uniform so as to arrange that for the reason you give in post #6.

 

[vcsharp2003]

Because when the rod is removed the (negative) charges on the sphere will redistribute to be uniform so as to arrange that for the reason you give in post #6.

Did you mean the reason I gave in post #9?

 

[haruspex]

Did you mean the reason I gave in post #9?

No, I meant the first half of post #6.
I'm not sure I understand post #9. It seems to be suggesting a mechanism by which charges would be stably distributed inside the sphere. That won't happen. At all stages of the sequence the charges are on the surface only.