- #1
Diracobama2181
- 75
- 2
- Homework Statement
- A sphere of radius R carries a charge q that is distributed uniformly over the
surface of the sphere. The sphere spins at constant angular velocity ω.
(a) Calculate the magnetic moment m of this sphere and determine the leading
term of the multipole expansion of the vector potential A at large distances
r : i.e., r >> R.
(b) Calculate the interaction energy between this sphere and an infinite straight
wire carrying a steady current I. The angle between the direction of the
current and the angular velocity vector ω is θ while the distance ρ between the wire and the sphere is much greater than the sphere radius: i.e.,
ρ >>R. Consider the case when ω is perpendicular to ˆe_ρ.
- Relevant Equations
- $$m=\int I da$$
$$A_{dip}=\frac{\mu_0}{4\pi}\frac{msin\theta}{r^2}$$ (from Griffiths)
Part a was not much of a problem. I got that $$m=QR\omega \hat{z}$$. From that, I get $$A_{dip}=\frac{\mu_0}{4\pi}\frac{QR\omega}{r^2}\hat{\phi}$$ (using $$\theta=\frac{pi}{2}$$.
My problem occurs in part b. I know there is a potential energy relation for two dipoles, but what would I use for a dipole and a wire? More specifically, is there a relationship between A and the interaction energy? Any suggestions or further readings on this subject would be appreciated. Thank you.
My problem occurs in part b. I know there is a potential energy relation for two dipoles, but what would I use for a dipole and a wire? More specifically, is there a relationship between A and the interaction energy? Any suggestions or further readings on this subject would be appreciated. Thank you.