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## Homework Statement

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).

## Homework Equations

ρ=Q/V

p=Ʃq_i(r_i-r)

E_sphere=Qr/4piεR^2 for r<R

superposition principle

## 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)