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

In summary, the conversation is about a student struggling with a question on a fields exam, particularly part a) and b) which involve calculating the energy density and electric field for a long cylindrical co-axial capacitor with varying dielectric constant. The student discusses their approach and asks for help with understanding the contribution of the inner conductor to the electric field.
  • #1
Beer-monster
296
0
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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  • #2
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?
 
  • #3
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?
 
  • #4
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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1. What is the purpose of solving a Fields Exam Challenge?

The purpose of solving a Fields Exam Challenge is to demonstrate an understanding of the principles and equations used to calculate electric field and energy density in a given system. It also allows for the application of these concepts to real-world scenarios and the ability to solve complex problems.

2. How do you calculate the electric field in a given system?

The electric field in a given system can be calculated using the equation E = kQ/r^2, where E is the electric field, k is the Coulomb's constant, Q is the charge of the object producing the field, and r is the distance from the object.

3. What is energy density and how is it calculated?

Energy density is the amount of energy stored per unit volume in a system. It is calculated by dividing the total energy by the volume of the system, or by using the equation u = 1/2ε0E^2, where u is the energy density, ε0 is the permittivity of free space, and E is the electric field.

4. What are some common challenges when solving a Fields Exam Challenge?

Some common challenges when solving a Fields Exam Challenge include correctly identifying and using the appropriate equations, understanding the units and conversions involved, and accurately representing the direction and magnitude of the electric field.

5. How can I improve my skills in solving Fields Exam Challenges?

To improve your skills in solving Fields Exam Challenges, it is important to practice regularly and familiarize yourself with different scenarios and variations of the equations. It can also be helpful to seek guidance from a teacher or tutor, and to break down complex problems into smaller, more manageable steps.

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