I worked out the multipole expansion for the vector potential of a line current because that's the most common type, and in some respects the easiest to handle. For a volume current J:

(a) Write down the multipole expansion, analogous to Eq. 5.78.

(b) Write down the monopole potential, and prove that it vanishes.

(c) Using Eqs. 1.107 and 5.84, show that the dipole moment can be written

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m = f(r x J) dt. (5.91)