Do the Maxwell equations definitively rule out the existence of magnetic monopoles?

Question

Gauss's law for magnetism, $\nabla \cdot \mathbf {B} =0$, is most directly interpreted as a sort of Kirchhoff's current law for magnetism, stating that while magnetic fields can be drawn between points (dipoles), they can't spring automagnetically (bad joke?) from single points. In other words, no monopoles.

And yet massive work is done on trying to "find the elusive magnetic monopole", notably recently at the London Centre for Nanotechnology. To ask the question, why the uncertainty? Considering the Maxwell equations and the Lorentz force law form the core of basically all of our models of electromagnetism, why the intensive search for something the mathematics say isn't there?

Answer 1

Because Maxwell's laws are empirical. We construct them by joining up what we know (Gauss law, Faraday-Lenz's law, and so on).

Ampère's law needed to be modified in order to solve some problems (Ampère-Maxwell's law aftwerwards), and that's what makes light possible.

So why couldn't we wrong? What if someday we find something so that $\vec{\nabla}\cdot \vec{B}\neq0$ anymore? Then monopoles would exist. It wouldn't be the first time we discover new things that change everything.

In short, mathematics say there aren't monopoles, but we did build that math.

Answer 2

Your thinking is the wrong way around. The direction is not: Maxwell's laws say that there are no monopoles, so there are none.

The correct direction is: There are no monopoles, so Maxwell's laws say that there are none.

To elaborate, since no experiment has ever produced reproducible magnetic monopoles, we have incorporated the empirical fact that there are no such monopoles into our theory. And that theory can then predict experimental outcomes assuming the observations they are built on are correct.

Searching for magnetic monopoles is a way to test wether our previous observations are indeed correct. As far as we can tell today, they are correct. But if they weren't, we'd like to know! So we can correct Maxwell's laws. That's why people are still testing.

Answer 3

If magnetic monopoles existed, Maxwell's equations would have to be extended and that would be a big break for the discoverer. The reward is high but the chances are low. There is as yet no indication at all that magnetic monopoles exist.

In condensed matter quasiparticles have been demonstrated that may be considered magnetic monopoles.

Answer 4

Question is on you, how magnetic field is produced from current if there is no source of magnetic field. If you think that Ampere's law accounted for producing magnetic field then not only you, whole science community is wrong.

See, Ampere's law $\nabla\times\mathbf{B=\mu J}$, is not about electric current producing magnetic field that curls around flow of charge, it is opposite of that. It says that curl of magnetic field produces current which we mistakenly identified as Faraday's law of induction $\nabla\times\mathbf{E=-\partial B/\partial} t$ for emf generation by changing magnetic field.

See from Lorentz's force law, $\mathbf{F}=q\,\mathbf{v\times B}$, mechanical force cause magnetic field to curl and they form rotation plane of the conductor and thus induced current flow perpendicular to the plane of rotation.

So question is about source of magnetic monopole, this is seen in any loop of current, a solenoid or an electromagnet. This shows that divergence of magnetic field exists, $\nabla\cdot\mathbf{B}=-\mu\,\mathbf{J/r}$. On solving this we get, $\mathbf{B}=\dfrac{\mu\,i}{2\,\pi\,r\,l}$.

For a loop having $N$ turns, $l$ distance long and inductance $L$, $\,\mathbf{B}=\dfrac{\mu\,N\,i}{2\,\pi\,r}=\dfrac{L\,N\,i}{2\,\pi\,r\,l}$

Below is the picture of summary of the above idea by chatGPT that, how curl of magnetic field can produce current as curl of current in a loop produces magnetic field. This shows that magnetic field has divergence and thus monopoles as a source.