Multipole expansion for Magnetic Vector Potential

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Multipole expansion for Magnetic Vector Potential

Post by blackcat007 » Mon Jun 14, 2010 9:06 pm

I am studying Griffiths and I have the following doubt. For the multipole expansion of the magnetic vector potential Griffiths simply writes the formula for a loop of wire and then ofcourse $$\oint d\mathbf{l}$$ is 0 around a loop (eq 5.78 and 5.80) and then he concludes that there is no monopole, but think about a finite segment of line charge that is moving at v(<<c) in that case the integral will not be a closed one, how do we then account for the fact that there are no monopoles then? Also why has he put a closed integral in the first place?

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Re: Multipole expansion for Magnetic Vector Potential

Post by twistor » Tue Jun 15, 2010 2:32 pm

You are probably overthinking this.

The Griffiths derivation must include the closed loop integral by the setup of the problem.

In the case of a line charge, think of the magnetic field produced. It must curl around the line charge and close in on itself. Thus, like all magnetic fields, the field produced has 0 divergence and therefore no monopole moment.

I don't have Griffiths in front of me but I'm sure if you work it out for a line charge you will conclude, if you do it correctly, that there is no monopole term.

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