Would magnetic monopoles violate the conservation of energy?

[agent_509]

My guess is that they don't since it's thought that they might be possible, but here's why I'm asking this question:

in the magnetic field created by a magnetic dipole, couldn't you place a magnetic monopole into the field (make sure it stays on the set path) and have it circulate continuously? I uploaded a picture, sorry for the crummy drawing, I just threw it together real quick in paint.

Anyways, why wouldn't this work if a monopole existed?

thanks for the help

Edit: sorry just realized this probably shouldn't have gone in classical physics seeing as maxwell's equations predict magnetic monopoles don't exist. At least it's still part of electromagnetism. Sorry.

 

[jtbell]

I haven't thought this out fully, but you might try replacing "magnetic monopole" with "electric charge," and "magnetic dipole" with "electric dipole," and considering whether it's possible for the electric charge to circulate continuously.

 

[agent_509]

Except that an electric dipole isn't a true dipole. There isn't a complete circulation. If you placed a positive charge just next to the positive end of the electric dipole, it would follow a path to the negative end of the dipole, but would be stuck there. An electric dipole has a source and sink point, whereas a magnetic dipole has no individual sources for its field.

 

[espen180]

You have to consider not only the work done on the charge, but also by the charge on the circuit, and the induced emf in the circuit, assuming you are using a superconducting circuit as your magnetic dipole.

At least if the charge is strong enough, it should be repelled from the circuit before it enters it, since the increased magnetic flux through the circuit is compensated by an equally strong induced outward flux. Then it might oscillate to and fro one end of the circuit indefinately. I donæt know what might happen if the charge is not strong enough to cause a reversal of the current.

 

[Dickfore]

If magnetic monopoles existed, then we would need to modify Gauss's Law for magnetic fields. Namely, in the presence of MMs, the magnetic flux through a closed Gaussian surface is no longer zero. Mathematically, we have:
<br /> \oint{\left(\mathbf{B} \cdot \hat{n}\right) \, da} = \mu_0 \, Q_M<br />
or, in a differential form:
<br /> \mathrm{div} \mathbf{B} = \mu_0 \, \rho_m<br />
Now the magnetic field is no longer solenoidal (has divergence zero everywhere). It can be proven mathematically that the necessary condition for the existence of closed field lines in a stationary vector field is that the field be solenoidal. Thus, just like electrostatic fields, you no longer have closed magnetic field lines.

 

[chrisbaird]

The picture is more complex than that. Charged particles have mass and therefore inertia. They don't follow field lines because their inertia draws them away. Put another way, a force causes the particle to accelerate and gain speed, which also must be accounted for when figuring out its trajectory in addition to the force at each point.

Also, an electrically charge particle that accelerates, including one that is just radially accelerating (i.e. curving its path) emits electromagnetic radiation. A magnetic monopole that is following a curved path would also emit electromagnetic radiation.

Lastly, work is net force times net displacement. So if an object goes somewhere, then returns to its original location, there is no net displacement, therefore no net work is done and no energy is expended. That is the reason (neglecting friction) a motionless child can swing forever back and forth on a swingset without needing to plug in the swingset.

 

[agent_509]

If magnetic monopoles existed, then we would need to modify Gauss's Law for magnetic fields. Namely, in the presence of MMs, the magnetic flux through a closed Gaussian surface is no longer zero. Mathematically, we have:
<br /> \oint{\left(\mathbf{B} \cdot \hat{n}\right) \, da} = \mu_0 \, Q_M<br />
or, in a differential form:
<br /> \mathrm{div} \mathbf{B} = \mu_0 \, \rho_m<br />
Now the magnetic field is no longer solenoidal (has divergence zero everywhere). It can be proven mathematically that the necessary condition for the existence of closed field lines in a stationary vector field is that the field be solenoidal. Thus, just like electrostatic fields, you no longer have closed magnetic field lines.

That's the answer I was looking for, thanks!

 

[Khashishi]

Eh, I don't think Dickfore's answer is sufficient. You can still have a solenoidal field if magnetic monopoles exist. Just don't use them. If rho_m = 0 then it just reduces to div B = 0.

In the electric field case, we can generate an EMF with a changing magnetic field. This produces a looped electric field which will cause electric charges to accelerate in a loop. This is just a generator. It takes energy to generate the changing magnetic field.

Let's consider the magnetic case now. If the magnetic field comes from a current, then it's clear that the energy needed to accelerate the magnetic charge needs to come out of the current. But, in the case of a permanent magnet, the magnetic field comes from aligned spins in the ferromagnetic material. I suppose the monopole will spiral into a point on the side of the magnet, where the magnetic field inside the magnet points north, and the field outside the magnet points south. And the monopole will stay there.

 

[Dickfore]

Eh, I don't think Dickfore's answer is sufficient. You can still have a solenoidal field if magnetic monopoles exist. Just don't use them. If rho_m = 0 then it just reduces to div B

So, there is no monopole that circulates.

 

[Khashishi]

Well, I don't think the monopole feels its own field. In any case, the monopole's field can be treated as small compared to the magnet's field.

 

[Dickfore]

Can you keep the field due to the monopole much smaller than the external field as you move arbitrarily close to it?

 

[VortexLattice]

First of all, we know right now that it doesn't break conservation of energy. Definitely not. That they would fit in with current theories is further proof.

But that doesn't really answer your question. A magnetic monopole is basically like either the N or the S side of the magnet, but with no corresponding S or N that must accompany it. You're correct, if you placed, say, in your drawing, a N along one of the field lines, it should follow that field line all the way to the S. But once there, why would it continue? It's basically exactly like jtbell said, I think. Either way, it doesn't violate it.

 

[The_Duck]

Can you keep the field due to the monopole much smaller than the external field as you move arbitrarily close to it?

I think this is irrelevant. If we take a point where the magnetic field points in some direction, and then place a magnetic monopole at that point, the magnetic charge will feel a force in the direction the field originally pointed, no matter that the field near the monopole now looks very different. This is true at all points along the original closed magnetic field lines, so the monopole will want to circulate around the original field lines. It doesn't matter that when you include the field of the monopole, the field lines don't form loops.

I think the answers citing the work the current must do to maintain the magnetic field are correct.

Consider the following variation of the original situation. We run a current through an infinitely long straight wire to set up a magnetic field that circles around the wire. Now we place a ring of magnetic monopoles around the wire. The ring of monopoles will start rotating with the magnetic field. This rotating ring of magnetic monopoles constitutes a loop of magnetic current. Just as a loop of electric current generates a magnetic field, the loop of magnetic current generates an electric field. The original electric current points opposite this new electric field and so additional work has to be done to maintain the original electric current. This additional work is exactly equal to the energy gained by the accelerating ring of magnetic monopoles (at least, I assume it is; I haven't done the calculation).

 

[Hassan2]

You have to consider not only the work done on the charge, but also by the charge on the circuit, and the induced emf in the circuit, assuming you are using a superconducting circuit as your magnetic dipole.

At least if the charge is strong enough, it should be repelled from the circuit before it enters it, since the increased magnetic flux through the circuit is compensated by an equally strong induced outward flux. Then it might oscillate to and fro one end of the circuit indefinately. I donæt know what might happen if the charge is not strong enough to cause a reversal of the current.

Your explanation is convincing, however the strength or the velocity of the charge does not matter. At the time the monopole passes through the center, the flux becomes discontinuous (because of a sudden change in direction of flux lines) which results in a high induced current which pull the charge back.

 

[Dickfore]

I think this is irrelevant. If we take a point where the magnetic field points in some direction, and then place a magnetic monopole at that point, the magnetic charge will feel a force in the direction the field originally pointed, no matter that the field near the monopole now looks very different.

Prove this statement.

This is true at all points along the original closed magnetic field lines,...

But there is at least one field line that ends on the monopole itself. Therefore, the rest of your argument is dubious.

I think the answers citing the work the current must do to maintain the magnetic field are correct.

Actually, the moving magnetic monopole induces a varying electric field (just as a moving electric charge induces a variable magnetic field) in the coil, that needs to be compensated by the emf source so that the current in the coil remains constant.

 

[The_Duck]

Prove this statement.

But this is exactly the same as the statement in electrostatics that if I place an electric charge in an electric field, the electric charge will feel a force in the direction of the original field, no matter that the field close to the point charge now looks very different. Do you dispute this?!

But there is at least one field line that ends on the monopole itself. Therefore, the rest of your argument is dubious.

I don't understand your objection here; I'm arguing that, as Khashishi said, we ignore the field of the monopole in computing the force on the monopole, and look at the original field into which we are placing the monopole. In the original magnetic field, field lines never end.

Actually, the moving magnetic monopole induces a varying electric field (just as a moving electric charge induces a variable magnetic field) in the coil, that needs to be compensated by the emf source so that the current in the coil remains constant.

Here we are in perfect agreement and merely using different words. Perhaps the same is true above.

 

[vanhees71]

Since one can write down a Lagrangian for em. fields, electric and magnetic charges that is not explicitly dependent on time, due to Noether's theorem the energy-conservation law still holds true.

The original papers by Dirac are a very nice read!

P. A. M. Dirac. Quantised Singularities in the Electromagnetic Field. Proc. Roy. Soc. A, 133:60, 1931.
http://www.jstor.org/stable/95639

P. A. M. Dirac. Theory of Magnetic Poles. Phys. Rev., 74:817, 1948.
http://link.aps.org/doi/10.1103/PhysRev.74.817