1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Potenital of a cylindrical, linear dialectric in a E-Field

  1. Oct 28, 2016 #1
    1. The problem statement, all variables and given/known data
    A very long cylinder of linear dialectric material is placed in an otherwise uniform electric field ##E_0##. Find the resulting field within the cylinder. (The radius is ##a##, the susceptibility ##X_e##, and the axis is perpendicular to ##E_0##)

    2. Relevant equations
    Boundary conditions
    ##V_i = V_o## at ##s=a##

    ##\epsilon \frac{\partial V_i}{\partial n} = \epsilon_0 \frac {\partial V_o}{\partial n}## because there is no free charge

    ##V = -E_0*s*cos(\theta)## for ##r>>a##

    3. The attempt at a solution
    I've gotten this far

    ##\epsilon_r[-na^{-n-1}* (Acos(n*\theta) + Bsin(n*\theta))] = na^{n-1} * (Ccos(n*\theta) + Dsin(n*\theta)) -E_0cos(\theta)##

    where n is what I'm summing over. I'm unsure of how to restrict the constants of the trig functions to reduce the problem.
     
  2. jcsd
  3. Oct 28, 2016 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    This equation isn't correct, as written. There should be a summation over ##n## on each side (except for the term ##-E_0 \cos\theta ~##, which is not summed over ##n##).

    For each value of ##n##, the coefficient of ##\cos(n \theta)## on the left side must match the coefficient of ##\cos(n \theta)## on the right side. Similarly for ##\sin(n \theta)##.

    So, what condition on the coefficients do you get when matching ##\cos(n \theta)## for the special case of ##n = 1##? What condition on the coefficients do you get when matching ##\cos(n \theta)## for ##n \neq 1##?

    Similarly, get equations from matching ##\sin (n\theta)##.

    There is also another boundary condition on ##V## (which you stated in the relevant equations section) that you need to invoke where you can again match coefficients on each side.
     
    Last edited: Oct 28, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Potenital of a cylindrical, linear dialectric in a E-Field
Loading...