Short version of my question:
Do dipole currents cause fields? I think currents of aligned magnetic dipoles cause an electric field, but I don't know how to calculate this field except in the simplest of cases. I'd like to know how!
Full version of my question:
Suppose I have a wire (or pipe) that carries a steady current of particles, and the wire is shaped like a sine curve in the $X$–$Y$ plane with spatial period $P$ and amplitude $A$. I want to calculate the electric and magnetic fields produced by the current in the wire.
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If the flowing particles are charged, I can use the Biot-Savart law to write an integral and (at least numerically) calculate the resulting magnetic field. In the limit where $A=0$, I can check my results against a Lorentz-boost calculation, and see that the two methods agree.
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If the flowing particles are electric dipoles pointing in the $Z$-direction, things are more complicated, but I can still use the Biot-Savart law, adding up the magnetic fields from two equal, opposite, displaced sinusoidal electric currents to get a net magnetic field, albeit one that falls off much more quickly than in the previous case. I can still check my results against the Lorentz boost in the $A=0$ limit, and find that the two methods agree.
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However, if the flowing particles are magnetic dipoles pointing in the $Z$-direction, I don't know how to calculate their electric fields except in the $A=0$ limit. In this limit, I can use a Lorentz boost argument to show that an electric field exists, but I don't know of any equivalent to the Biot-Savart law for currents of magnetic dipoles. I would be very surprised if the electric field vanished in the $A>0$ case. How do I proceed? Is this covered in elementary textbooks, and I've missed it somehow?
I'm tempted to model this third case in a similar fashion to case 2, as a fictitious pair of equal and opposite "magnetic currents", and use an electric version of Biot-Savart. This produces the correct result in the $A=0$ case, and seems reasonable in the $A>0$ case, but seems to have no basis in any textbook or reference I can find. What am I missing?
Caveats:
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If at all possible, please base your answer on credible textbooks or peer-reviewed papers. For example, if you think Maxwell's equations is missing a $\vec{v} \times \vec{M}$ term, I'd very much like to see an external link to back it up.
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This may be obvious from the 'spirit' of the question, but I'm not particularly attached to the sinusoidal shape of the current-carrying wire. Any nontrivial shape of the wire that precludes a Lorentz boost is equally interesting to me, so go ahead and change the wire's shape if it makes the math easier.
Best Answer
Since there seems to have been some doubt about whether the electric field of a boosted magnetic dipole is nonzero, this paper by Hnizdo (what a great name!) may be helpful. Section 3 explicitly calculates the field. This is all tangentially related to the Mansuripur paradox as well.
I think dj_mummy's technique works, but it has the disadvantage that it requires you to do a calculation for some finite $l$ and the take the limit $l\rightarrow0$. Here's a different technique, which avoids that.
The vector potential due to a magnetic dipole at rest is $A^\mu=(\phi,\textbf{A})$, with $\phi=0$ and $\textbf{A}=\textbf{m}\times\hat{\textbf{r}}/r^2$. Do a Lorentz boost on this vector, and you get a potential $A^{\mu'}$ for a moving dipole. In fact, you only care about the timelike component of this, so you don't need to compute the rest. Integrate this over all the dipoles (each with its own position and velocity vector), and then take the gradient to get the electric field. Note that although the electric field is $-\nabla\phi-\partial\textbf{A}/\partial t$, the second term can be neglected; the integrated value of $\textbf{A}$ is constant in the $\mu'$ frame, since the dipole current is static in that frame.
Another way to approach this, suggested by Art Brown in a comment below, goes like this. Hnizdo shows that to a sufficient approximation, and ignoring some subtleties related to the definition of multipoles, we can take a magnetic dipole $\textbf{m}$ to have electrical properties, in the lab frame, characterized by an electric dipole moment $\textbf{p}'=\textbf{v}\times \textbf{m}$ (in units with $c=1$). This makes the whole problem look directly analogous to the idea of developing the Biot-Savart law by assembling a collection of magnetic dipoles, so although I haven't worked it out in detail, it sounds like you can get something that's an exact analog of the Biot-Savart law.