I'd type this out but there's a bit too much formulae.
It's problem 2. I'm just wondering if my solution is correct.
Thanks in advance!
The Attempt at a Solution
In this problem I'm taking Ro = |xo - x'|
The volume in question is charge-free so the charge density, ρ(x'), is zero so the first term on the right hand side of the potential vanishes.
Also, ∂/∂n(1/Ro) = -1/R2o
Substituting this into the potential function gives the required result.
For the second part of 2 we use the divergence theorem (which I won't state here due to my lack of latex skills) as told.
We know that ∂[itex]\Phi[/itex]/∂n = ∇[itex]\Phi[/itex].n and from the definition of the electric field E we end up with -E.n.
When this is used in the divergence theorem we end up with the volume integral of ∇.E which is equal to ρ/ε which vanishes in a charge free volume.
We now have the required expression.
(Haven't gotten to part 3 yet, will be posted soon.)
I know I haven't explained everything in a great way, but it's a lot easier on paper than it is online to write out loads of partials and surface integrals.