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Is magnetic field conservative in nature?

Magnetic field lines do go in closed paths but that's not the definition of conservative.

Rather, a field is conservative when the force on a test particle moving around any closed path does no net work. But magnetic fields only act on moving charges, and at right angles to the motion, so the work is always zero and the concept doesn't properly apply.

Also, if there were magnetic monopoles, they would try to follow the magnetic field the way electric charges try to follow the electric field lines.

Now we consider the magnetic field due to magnetic monopoles, which is going to be conservative.

So Is there a precise answer? or the nature of the magnetic field depends on the way you produce it? If the latter is the case then, is the magnetic field created by monopoles conservative?

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A field $\mathbf{F}$ is conservative if and only if $\nabla \times \mathbf{F}=0$. From Maxwell's equations we know that

$$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\epsilon_0 \frac{\partial\mathbf{E}}{\partial t}. $$

Hence, the magnetic field is only conservative in the absence of free currents and time varying electric fields.

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  • $\begingroup$ I like this answer, but the Maxwell-Ampère equation that you quote applies to each point. Do we insist that for a field to be conservative there are no currents or time-varying electric fields at $any$ point? I'm considering the field around a long wire carrying a steady current. Curl B is zero everywhere except in the wire, because J is zero everywhere except in the wire. Yet we'd say, wouldn't we, that the magnetic field is non-conservative? $\endgroup$ Jul 14, 2018 at 16:44
  • $\begingroup$ Still, the magnetic field created by monopoles is conservative, Right? $\endgroup$ Jul 15, 2018 at 4:51
  • $\begingroup$ "Do we insist that for a field to be conservative there are no currents or time-varying electric fields at any point?" Yes. $\endgroup$
    – Chris
    Aug 16, 2018 at 15:12
  • $\begingroup$ @NKKhiwaal As far as we know today there exist no magnetic monopoles since the magnetic field is sourcefree. $\endgroup$ Aug 20, 2018 at 10:55
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    $\begingroup$ @Diracology Why don't you elaborate on your statement? What is the point of making a vague statement contradicting the core point of an answer without providing examples to support your stand nor providing an alternate answer to OP's question? $\endgroup$ Jul 17, 2022 at 8:50

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