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Homework Help: Stumped on E&M problem

  1. Dec 10, 2007 #1
    1. The problem statement, all variables and given/known data
    infinitely long conducting cylinder (hollow, so outter radius = b and inner radius = a) which is grounded is in a constant E-field (E=Eo in direction of cylinders' axis) has a dielectric substand "painted" on it, with dielectric constant c and thickness D. Find the electric field everywhere!

    2. Relevant equations
    Well, for dielectrics the usual BC's are:
    E[parallel](inside) = E[parallel](outside)
    D[perpendicular](inside) - D[perpendicular](outside) = sigma <-- charge density

    Also, the general solution for potential in cylindrical coords is:
    V = Ao + Bo*ln(r) + [tex]\sum(An*r^n +Bn*r^(^-^n^))*(Cn*cos(n*phi)+Dn*sin(n*phi)[/tex]

    BC for infinite cylinder:
    dV/dz = 0
    V = -Eo*r*cos(phi) + constant, as r-->infinity (due to constant E-field.)

    3. The attempt at a solution
    I'm having some serious trouble with this problem. One thing I do know, however, is that all of the Dn*sin(n*phi) terms contribute nothing by symmetry and so we can kiss the Dn's goodbye. I think the BC's I stated above are correct, but I'm having problems using them. Other questions I have are: i) how does the dielectric effect V, and, ii)despite being grounded, wouldn't the external field effect the cylinder hence making it act like a dipole?

    Finally, can we assume the E-field inside is zero since it is a conductor? Or does the dielectric and external E-field change this? The E-field in the region a<r<b should be zero too, shouldn't it?

    Any help would be great!
     
  2. jcsd
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