How Does the Second Uniqueness Theorem Determine the Electric Field in a Volume?

[pardesi]

it states that in a given volume V surrounded by conductors or for that matter infinity if the charge density \rho and the charge on each conductor is fixed then the electric field is uniquely determined in that volume V

Can someone use this find the field in certain situations.
For Example consider this classical situation where in an uncharged conductor has a cavity of arbitrary shape inside it which has a point charge q inside it .The question is to find the net field outside it .
Ofcourse the answer is shielding by the metallic sphere ?
Can someone prove this using the uniqueness theorem .I have a proof in mind but i am unsure of it?

 

[pam]

What does "net field" mean?
The field outside the conductor will be the same as that outside a solid conductor with total surface charge q. The field outside such a conductor depends on its shape.
By Gauss's law, the surface integral of E is known.

 

[pardesi]

What does "net field" mean?
The field outside the conductor will be the same as that outside a solid conductor with total surface charge q.

this is what i am asking u to prove.
also that the charge q is uniformly distributed

 

[pam]

The surface charge on the conductor has to be q, by conservation of charge.
The surface charge will not be uniform, unless the outer surface is spherical.

 

[pardesi]

it is a sphere ...
even that doesn't 'prove' that the charge is uniform

 

[pam]

The title of your question was "uniqueness theorem". Use it.
If a spherically symmetry E outside the conductor satisfies all BC, then it is the unique solution. QED.