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.