Charge inside a cavity in a conductor My teacher told that when a charge is placed inside an arbitrarily shaped cavity in a spherical conductor, the charge occurring on the outer surface is distributed uniformly, and this will imply that the field due to the cavity and charge will be zero at all points outside it but I can't figure out why. I think that this will imply that even if the conductor is not spherical, the outer charge will attain the same charge configuration as that of a conductor of same size and shape but with no cavity.
 My teacher gave the following argument to prove it.Since all the gaussian surfaces which enclose the cavity and inside the conductor have zero flux , which is possible only if the field due to the cavity and the conductor is zero. But how to prove  this mathematically?
I thought that if the conductor were of bigger size but same shape, then the distribution of charges in the inner surface of cavity will not change, so we can assume the conductor to be of infinite size, and the result will follow. i don't know if this is right. I also can't justify that the distribution will not change. Please help.
 A: 
I think that this will imply that even if the conductor is not spherical, the outer charge will attain the same charge configuration as that of a conductor of same size and shape but with no cavity.

This is correct!
Now some step by step explanations.


*

*If you take any closed surface and an electric charge $q$, then the total flux of electric field through the surface will be either 0 (if the charge is "outside"), or $q/\epsilon_0$ (if the charge is "inside"). Proof of this statement is a little bit tedious, but only compared to the following steps, which require no calculations at all.

*Electric field inside a conductor must be zero. Otherwise the free charges in conductor will redistribute because of the electric field.

*From statement 1. follows that total flux through any closed surface depends on total charge inside the surface.

*There can be no charge density inside the conductor. Suppose there is some charge density - let's consider small surface around in this part of conductor. There is some charge inside => there must be non-zero flux through the surface. But the flux must be zero because the electric field is zero at any point of the surface, because it's inside the conductor. That means that all the charges must be located not inside, but on the surface on conductor.

*Now we have a conductor, cavity and  a charge inside cavity. There will be some charges the surfaces of the conductor. So, there are 3 types of charges: the charge inside cavity, charges on inner surface, charges on outer surface. Electric field at any point of space is produced by (only) these charges.

*Let's take a surface inside the conductor, which encloses the cavity. Flux is zero => total charge inside is zero => if the point charge inside cavity is $q$ than total charge on inner surface is $-q$, total charge on outer surface is $q$.

*Let's remove the cavity together with all the charges it contains. Now we have a conductor, total charge of the conductor is $q$, the charges are distributed on the surface of the conductor in such a way, that the electric field in any point inside the conductor is zero. Let's remember the distribution of charges.

*Let's take a space completely filled with conductor, make cavity in it and put $q$ inside the cavity. The will appear $-q$ charge on the surface, and the charge on the surface will distribute in such a way, that the electric field in any point outside the cavity is zero. Let's remember this distribution of charges as well. You see, this is a distribution that produces no electric field "outside" and hence does not affect outside charges at all!

*Now let's combine previously remembered distributions from steps 7. and 8. We'll see, that charges on outer surface do not affect the inner charges, and inner charges would not affect outer charges. So, this "combined" distribution of charges is also possible!

*We have found a stable distribution of charges. "The same configuration of charges on the outer surface is possible no matter what is the shape and location of the cavity and position of the charge inside cavity as long as the total charge of the conductor is $q$". I thinks it's cool, but strictly speaking we do not know yet if the charges would actually distribute this way. May be there is some other stable distribution? Well, there is a theorem that there is only one way charges can distribute on the surface of conductors. Proof is not very difficult, I can go into it if you are interested.
UPDATE. Some clarification about "combining" charges distributions in step 9.
In step 7 we found some distribution of charges on the outer surface, and this distribution is in equilibrium. That means that if we calculate the sum of forces acting on any small charge in the system from all other charges in the system the result would be orthogonal to the surface (remember, that all extra charges accumulate on the surfaces). Would it be not orthogonal to surface the charge would be "dragged" along the surface and redistribute somehow. So, we remember this distribution.
In part 8 we create and remember another distribution of charges, also in equilibrium.
And now we combine these distributions. We artificially create a new distribution of charges, we fix all the elementary charges in place and do not allow them to move. Now we look at what we have and understand that there is no need to hold the charges. We let them go and they do not have to redistribute. Because charges from step 7 do not affect charges from step 8. The total electric field produced by charges from step 7 create zero electric field in the regions there charges from step 8 are located. And other way round!
