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References (e.g. this one) usually say that the force on a magnetic dipole in a magnetic field is

$$\vec{F} = \nabla\left(\vec{\mu} \cdot \vec{B}\right) $$

So consider a circular loop in the $xy$ plane with current going around it. Suppose the $B$ field is in the $z$ direction, but $\frac{\partial B}{\partial z} \neq 0$.

Then the formula for the force is non-zero, but the force by the Lorentz force law is zero.

I suppose the problem is that the $B$-field I described isn't physical, since is has non-zero divergence, but where does that break the derivation of the law for the force on a dipole?

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  • $\begingroup$ Though it's clear that your field configuration violates Maxwell's Equations, I'm having trouble seeing intuitively why this field isn't physical. Why couldn't you create a field like this with, for example, a solenoid whose resistance steadily increases (and current correspondingly decreases) along its length? $\endgroup$
    – probably_someone
    Commented Aug 23, 2018 at 16:37
  • $\begingroup$ @probably_someone two different resistors connected in series would still have the same current through them, or do you have something else in mind? $\endgroup$
    – hyportnex
    Commented Aug 23, 2018 at 17:09
  • $\begingroup$ What do you mean the Lorentz force law gives zero force? $\endgroup$
    – SuperCiocia
    Commented Aug 23, 2018 at 23:45

2 Answers 2

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Answer rewritten due to misunderstanding the quesiton.

The fact that the field you propose has a non-zero divergence breaks the derivation of the dipole force law at equations 10 through 12. The force on a wire loop parallel to the $x-y$ plane only occurs due to the transverse magnetic fields, namely $B_x$ and $B_y$ as seen in Equation 10. $$f_z = -|\mu|(\partial_yB_y + \partial_xB_x)$$ Then, by assuming that the field is divergence-less (Equation 11) $$\partial_xB_y + \partial_xB_x + \partial_zB_z = 0$$ we can simplify the force equation as (Equation 12) $$f_z = |\mu|(\partial_zB_z)$$ A magnetic field that has a gradient along the z-axis must have gradients along the perpendicular directions to remain divergence-less. These perpendicular fields provide the Lorentz force on the wire loop.

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  • $\begingroup$ "There seem to be two wire loops in your question." No. "The paragraph after that considers the wire loop that generates the magnetic field" No, I just imagined an external magnetic field and didn't discuss its source. "The Lorentz force is not zero in the reference article." I know but that's not what I'm asking. I'm asking about the situation described in my question. "Just because ∂Bz/∂z≠0 does not mean the divergence is non-zero. " Yes it does, because my question also said "Suppose the $B$-field is in the $z$-direction." $\endgroup$
    – Mark Eichenlaub
    Commented Aug 24, 2018 at 5:01
  • $\begingroup$ @MarkEichenlaub I've rewritten my answer now that I understand what you were going for. $\endgroup$
    – Mark H
    Commented Aug 24, 2018 at 8:57
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For the force acting on the current loop lying in the $xy$ plane to be non-zero, the magnetic field gradient should have a non-zero component in the $xy$ plane, so $\frac{\partial B}{\partial z} \neq 0$ condition is not sufficient.

Here is a relevant quote from your reference article, where the gradient of the magnetic field is assumed to be in $y$ direction:

If the gradient of B in the y direction is nonzero, then the forces shown in figure 2 will not cancel.

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