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Homework Help: Electromagnetic field theory

  1. Jan 31, 2014 #1


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    1. The problem statement, all variables and given/known data
    Consider a spherical dielectric shell so that ε = ε_0ε_r for a < r < b and ε = ε_0 for 0 < r < a. If a charge Q is placed at the center of the shell, find

    a) P for a < r < b
    b) ρ_pv for a < r b
    c) ρ_ps at r = a and r = b

    2. Relevant equations
    ρ_pv = -div(P)
    ρ_ps = P [itex]\cdot[/itex]a_n

    3. The attempt at a solution
    I've managed to solve a) with the answer
    Q/(4*pi*r^2) * (ε_r-1)/ε_r​
    which is correct (checked the answers).

    I've also solved b) which is ρ_pv = 0 since ρ_pv = -div(P).

    I have trouble to find ρ_ps at r = a and r = b. The answer states that
    -Q/(4*pi*a^2) * (ε_r-1)/ε_r and -Q/(4*pi*b^2) * (ε_r-1)/ε_r ​
    respectively. Where is that negative sign coming from?
  2. jcsd
  3. Jan 31, 2014 #2


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    On the r = a I would guess that we get a negative sign since we're having a unit vector a_r which goes outward from the origin. And since the normal to the surface when r = a is the -a_r I would guess that the negative sign when r = a is correct. But when r = b, the unit vector from P and a_r are booth positive, so why negative there?
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