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Charged Sphere with off-center cavity Electric field

  1. Jan 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider a sphere uniformly charged over volume, apart from a spherical
    off-center cavity. The charge density is ρ, radius of the sphere is a, radius of the cavity is
    b, and the distance between the centers is d, d < a-b. (a) Find the total charge and the
    dipole moment (with respect to the center of the large sphere) of this configuration. (b) Use
    superposition principle to find the electric field inside the cavity. (c) Show that far from the
    sphere the field is that of a charge plus dipole correction. Check that the charge and the
    dipole moment correspond to that of part (a).


    2. Relevant equations
    ρ=Q/V
    p=Ʃq_i(r_i-r)
    E_sphere=Qr/4piεR^2 for r<R
    superposition principle


    3. The attempt at a solution
    total charge I'm fairly certain is (4/3)piρ(a^3-b^3) just the large sphere minus the cavity.
    The dipole moment i attempted to use a sum p=q_a(0-0)+q_b(d-0) and got

    p=(4/3)pi*ρ*b^3*d (from the center of the cavity towards the center of the large sphere)

    for b) I tried to find the Electric field due to the large sphere ((4/3)piρa^3)*(r/4piεa^2) and the field from the small sphere ((4/3)piρb^3)*(r/4piεb^2) but im not sure what coordinate system i should be using, nor how to superimpose/sum the fields correctly.

    c) We were not taught nor can i find anything in the book about a dipole correction, so I'm lost for this part.

    (I wasnt sure if this should go in Advanced or Introductory physics, so it is in both, I will remove the other as soon as one is replied too, sorry)
     
  2. jcsd
  3. Jan 25, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    You can consider the hollow sphere as a full one + a small sphere with opposite charge, so you get zero charge inside the cavity. You can direct your system of coordinates any way, choose the most convenient: the centres of the spheres line up along one coordinate axis.
    According to this model, the dipole moment is the same as that of a point charge q at distance d from the centre of the big sphere. Take care with the sign: The dipole moment points from the negative charge to the positive one.
    The electric field of the hollow sphere is the sum of the full sphere + the small oppositely charged sphere at the place of the cavity. Check your work: The electric field at distance r depends on the charge enclosed in the sphere of radius r. If r<b the enclosed charge is less than the total charge of the sphere.

    ehild
     
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