Solving a Fields Exam Challenge: Calculating E Field & Energy Density

[Beer-monster]

Okay I've got a fields exam coming up so like a good boy I've been practising with past papers but there is this one question that is driving me batty

a) Consider a long cylindrical co-axial capacitor with inner conductor radius a, outer conductor radius b, and a dielectric constant that varies with cylindrical radius K(r). Show that for the energy density the dielectic to be constant, K(r) must equal k/r^2.

b) Given that the capacitor is charged to voltage V, determine the electric field E(r) as a expression of V, r, a and b.

Okay part a I can sort of do by calculating the E field based on Gauss' Law and subbing into the expression for energy density. However this approach requires that the charge density of the capacitor is constant throughout, which the question does not specify and seems a bit of a leap of faith.

part b I have no idea with, except it probably involves the boundary conditions of the E field and D.

Please help, this subject is starting to make quantum mechanics look easy

 

[turin]

... the charge density of the capacitor is constant throughout, which the question does not specify and seems a bit of a leap of faith.

If you mean the surface charge density on a given cylinder, then the symmetry of the capacitor ensures it.

b) Given that the capacitor is charged to voltage V, determine the electric field E(r) as a expression of V, r, a and b.

You can calculate the λ on the inner conductor from the V and C. Then, you can use the electric field for a line of charge in a dielectric to find the contribution from the inner conductor. Inside the capacitor and thus inside the outer conductor, what do you think the contribution to the E-field is and why?

 

[Beer-monster]

Is the contribution to the E-field from the inner conductor the electric field that radiates outwards from the cylinder. This it would be the component that is normal to the boundary, then could I calculate an expression for e-field based on the discontinuity expression of the normal displacement D1n-D2n=sigma?

 

[turin]

Is the contribution to the E-field from the inner conductor the electric field that radiates outwards from the cylinder.

Yes.

This it would be the component that is normal to the boundary, then could I calculate an expression for e-field based on the discontinuity expression of the normal displacement D1n-D2n=sigma?

What boundary? Both E and D terminate on a conductor (AFAIK), so I guess it's kind of a trivial boundary.