Charge and current distribution for magnetic dipole in motion

[Rocky Raccoon]

Homework Statement

Find rho(r,t) and j(r,t) for the following:

a) point charge q in uniform motion
b) electric dipole p in uniform motion
c) magnetic dipole m in uniform motion

Homework Equations

j(r,t) = v rho(r,t), so if we know rho, we know j

The Attempt at a Solution

a) is trivial - rho(r,t) = q delta (r - vt), where delta is Dirac's delta and v is the velocity of the charge

b) is also easy - rho(r,t) = - p . grad (delta(r - vt))

c) is where I don't know how to proceed.

I wish to solve the wave equation for the potentials A and phi but don't know what are the sources rho and j in the case of a magnetic dipole moment m.

 

[mark726]

Hello,

For the case of a magnetic dipole moment m in uniform motion, the sources for the potentials A and phi are the same as for an electric dipole in uniform motion. This is because the equations for the potentials A and phi are the same for both electric and magnetic dipoles, only the sources are different.

Therefore, for a magnetic dipole in uniform motion, the sources would be:

rho(r,t) = - m . grad (delta(r - vt))

j(r,t) = v rho(r,t) = - v m . grad (delta(r - vt))

I hope this helps! Let me know if you have any further questions.