# Homework Help: Charged Sphere with off-center cavity Electric field

1. Jan 24, 2012

### EnderTheGreat

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.