Charged Sphere with off-center cavity Electric field

[EnderTheGreat]

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 I am 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.

 

[mark726]

Thank you for your post. Let me address each part of your question separately.

a) You are correct in your calculation of the total charge. However, I believe your calculation for the dipole moment is incorrect. The dipole moment is the sum of all the individual charges multiplied by their respective distances from the origin. In this case, the origin is the center of the large sphere. So, the dipole moment should be p = q_a*d - q_b*(a-d) = (4/3)piρ*b^3*d - (4/3)piρ*a^3*(a-d).

b) To find the electric field inside the cavity, we can use the superposition principle. The electric field inside the cavity will be the sum of the electric fields from the large sphere and the small sphere. The electric field from the large sphere can be found using the equation E = (1/4piε)*(Q/r^2), where Q is the charge of the large sphere and r is the distance from the center of the large sphere. The electric field from the small sphere can be found using the equation E = (1/4piε)*(q/r^2), where q is the charge of the small sphere and r is the distance from the center of the small sphere. Since the electric field from the large sphere will be pointing away from the center of the large sphere and the electric field from the small sphere will be pointing towards the center of the small sphere, we can add these two fields together to get the total electric field inside the cavity.

c) The dipole correction refers to the fact that when we are far away from the sphere, the electric field can be approximated as the sum of the electric field from a point charge and the electric field from a dipole. The electric field from a point charge is given by E = (1/4piε)*(q/r^2), where q is the charge of the point charge and r is the distance from the point charge. The electric field from a dipole is given by E = (1/4piε)*(p/r^3), where p is the dipole moment and r is the distance from the center of the dipole. So, when we are far away from the sphere, we can use these equations to approximate the electric field. To check that the charge and dipole moment correspond to those found in part (a), we