Question

What is the force exerted on the dipole by the sphere a?

Answer 1

Have a point charge and a perfect dipole p⃗ p→ a distance rr away. Angle between p⃗ p→ and r^r^ is θθ. Want to find force on dipole.

I'm having more than a little difficulty identifying where I'm going wrong. If I do this problem in cartesian coordinates, I get the right answer, so apparently I am not understanding something about spherical coordinates.

We have F=qΔEF=qΔE for dipoles in a nonuniform electric field. If dd in dipole is small, then I can use

ΔE≈∇E⋅Δr⃗ ΔE≈∇E⋅Δr→

Below I derive the expression in spherical coordinates.

So, first of all,

E=q4πϵ0r2r^E=q4πϵ0r2r^

So

Er=q4πϵ0r2Er=q4πϵ0r2

and

ΔEr=∇Er⋅Δr

Hope it helps you bro