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Homework Help: Dipole in a dielectric medium

  1. Jul 25, 2011 #1
    1. The problem statement, all variables and given/known data
    A dipole p is situated at thecentre of a spherical cavity of radius a in an infiite medium of relative permitivity [itex] \epsilon_r [/itex]. show that the potential in the dielectric medium is the same as would be produced by a dipole p' immersed in a continuous dielectric, where

    [tex] p'=p\frac{3\epsilon_r}{2\epsilon_r +1} [/tex]

    and that the field strength inside the cavity is equal to that which the dipole would produce in the absence of the dielectric, plus a uniform field E

    [tex] E=\frac{2(\epsilon_r-1)}{2\epsilon_r + 1}\frac{p}{4\pi\epsilon_0a^3}. [/tex]

    2. Relevant equations

    3. The attempt at a solution
    I am not sure at all how to approach this question. I would like to say that I would use spherical harmonics but i am not sure how to apply them in this case.

    Would it be possible to say that at large distances

    [tex] V_2= -\frac{p\cos\theta}{4\pi\epsilon_0\epsilon_r r^2} [/tex]

    then to add then assume that outside the sphere that

    [tex] V_2= -\frac{p\cos\theta}{4\pi\epsilon_0\epsilon_r r^2} + \frac{A_2\cos\theta}{r^2} [/tex]

    and inisde the sphere that

    [tex] V_1= B_1 r \cos\theta + \frac{B_2\cos\theta}{r^2} [/tex]

    and then solve the problem using the boundary conditions for tangential E and perpendicular D?

    I am really unsure of how to solve this and any help will be greatly appreciated.
  2. jcsd
  3. Jul 25, 2011 #2
    Kind of. Since [itex]A_2[/itex] is unknown, you might as well just try the potential: [itex]V_2 = - \frac{p^{\prime} \cos\theta}{4\pi \epsilon_0 \epsilon_r r^2}[/itex], where [itex]p^{\prime}[/itex] is unknown and to be solved.

    The whole point of the problem is that at large [itex]r[/itex], the field looks like something due to some effective dipole moment [itex]p^{\prime}[/itex], whose value you are to find.

    [itex]D[/itex] is okay, but instead of using [itex]E[/itex], it is easier to use the condition that [itex]V[/itex] is continuous.

    So now you have two equations but three unknown: [itex]B_1[/itex], [itex]B_2[/itex], and [itex]p^{\prime}[/itex]. But one of them can be found by consider the limit [itex]r\to 0[/itex].
    Last edited: Jul 25, 2011
  4. Jul 26, 2011 #3
    Ahh ok that makes a lot more sense, thank you for the help.
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