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

AI Thread Summary
The discussion focuses on solving a challenging fields exam question about a cylindrical co-axial capacitor with a variable dielectric constant K(r). For part a, the user is attempting to show that K(r) must equal k/r^2 for the energy density to remain constant, relying on Gauss' Law but questioning the assumption of constant charge density. In part b, the user seeks to derive the electric field E(r) in terms of voltage V, inner radius a, and outer radius b, suggesting the need to consider boundary conditions and the contributions from the inner conductor. Participants emphasize the importance of understanding the electric field's behavior and the relationship between electric displacement D and surface charge density. The conversation highlights the complexities of the problem, particularly regarding assumptions about charge density and boundary conditions.
Beer-monster
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Okay I've got a fields exam coming up so like a good boy I've been practising with past papers :wink: but there is this one question that is driving me batty :confused:

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 :wink:
 
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Beer-monster said:
... 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.




Beer-monster said:
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?
 
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?
 
Beer-monster said:
Is the contribution to the E-field from the inner conductor the electric field that radiates outwards from the cylinder.
Yes.




Beer-monster said:
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.
 
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