More Magnetic Monopoles, With a Summery Hint of Maxwell’s Equations

It was pointed out to me that perhaps last time I went off too far into the theoretical setup and didn’t quite wrap up succinctly for you what exactly a magnetic monopole is. In short, a magnetic monopole would be a particle that carries magnetic charge, like how electrons and protons are carriers of electric charge. A bar magnet has two poles, and if you cut it in half, it still has two poles. If you keep cutting it in half and break it down as far as it will go, you will have a spinning electron which still has a “North” and a “South” pole. Whereas, in seeking the most simple possible configuration that produces an electric field, if you broke down a material as far as it would go you would have a single electron radiating a uniform electric field in all directions. This electric field wouldn’t pull objects toward it on one side and push objects away on the other like a dipole; it’s uniform in all directions (pictured here is the electric field of a positive point charge. An electron is a negative charge, so the direction of the field in reversed — pointing in toward the electron — but you get the idea). An electron is an example of an electric monopole. Similarly, a magnetic monopole, which is a magnetic charge, would have a uniform magnetic field radiating uniformly in all directions.

It’s not for the faint of heart, but for those willing to brave some math, I’ve got more for you on Maxwell’s equations, and how they would be symmetrical if a magnetic charge existed. Look at the pretty equations and skip to the summary just above the second set of equations if your eyes start glazing over. These are Maxwell’s equations for charges and electric and magnetic fields in a vacuum:

I’ve written these slightly differently than the standard form to emphasize the fact that electric charges create electric and magnetic fields. So the information about the fields is on the left-hand side, and the information about the charges is on the right-hand side. What these fields tell us is the nature of the magnetic and electric fields produced by charges. At least in theory, the strength and direction of the fields at any given point in space can be determined from any arbitrary set of charges or charged objects. In practice, trying to model complex geometries of charge distributions can be exceedingly tedious.

Before you get overwhelmed, let’s just look at the first two of Maxwell’s equations. The E represents the electric field and the B represents the magnetic field. On the right-hand side of the first equation, the Greek letter rho represents the electric charge distribution, while the epsilon with the zero subscript is a constant number that always stays the same, like pi or the square root of two. In examining equations with the goal of conceptual understanding in mind, the constants can be safely ignored. The right-hand side of the first equation is written in a generalized way so that the equation applies to any distribution of charge. For our purposes, it’s easier to imagine what the equations are saying if we decide that the information on that side of the equation represents a single point charge, such as a proton or electron. The triangle, called “del”, followed by a dot represents the way the electric field is spreading out from a charge. In the second equation, we have del dot B — so the spreading out of the magnetic field — and this is equal to zero. There is no magnetic charge from which a magnetic field radiates.

In the third equation, del followed by an X and an E, (spoken out loud as “del cross E”), represents the way the electric field is curling around the charge distribution. The funny d and dt coupled with the B represent how the magnetic field changes in strength and direction as time progresses. On the other side of the equation, we have zero, so there is no charge involved. A changing magnetic field actually induces and electric field. If you had just a static point charge — or even moving charges that are constant in their motion and quantity, and we shall see momentarily from the fourth equation — the magnetic field would not be changing. In fact, with a stationary charge distribution, as we saw from the second equation, no magnetic field at all would be produced. It follows that the electric field of a stationary charge (or unchanging electric current) does not curve. The electric field of a point charge only points straight outward or straight inward; it does not curl around.

In the final equation, del dot B represents the way the magnetic field is curling around. Then there’s the scary term involving Greek letters (which we can ignore today since they are constants), and another set of funny d’s, this time involving an E. Ignore the constants and what you have left is a term just like the one directly above it except with an E instead of a B. So this term represents the way the electric field is changing over time. On the right-hand side of the equation, we have another ignorable Greek letter, and a J. The J represents a distribution of electric current. Electric current is just made up of a bunch of moving point charges. If the electric field is not changing, then the second term on the left-hand side is zero, and we’re left with the statement that the way the magnetic field of a charge distribution curls is dependent on the way those charges are flowing in a current. If you have a current running through a straight wire, the magnetic field will curl around it in a circle, as shown here.

I’ll spare you a discussion of changing electric and magnetic fields, since that portion of Maxwell’s equations would be unchanged with the addition of magnetic charges. So, the long and short of it is, a magnetic field does not spread outward from an electric charge (just so that we’re clear, all discussion of “charges” up until this point has been a reference to “electric charges”, such as electrons and protons), but it will curl around the direction of flow of moving charges. The electric field does the opposite; it does not curl around the electric charges, but spreads uniformly outward from them.

And now (drum roll please), this is what Maxwell’s equations would have to be updated to look like if magnetic monopoles are discovered:

Here I’ve added subscripts to the rho’s and J’s to distinguish electric charges and currents from magnetic charges and currents. A magnetic charge, or monopole, is an entirely different object from those science words you no doubt have at least a passing familiarity with, like protons and electrons. None of the dozens of subatomic particles you’ve read about physicists discovering and debating carry this magnetic charge, as far as we have been able to detect. A magnetic charge would do exactly what an electric charge does in Maxwell’s equations as we know them. A magnetic field would radiate straight out from a magnetic point charge, but no magnetic field would curl around it. An electric field would curl around a moving magnetic charge or current, but would not radiate outward from the magnetic charge.

If you ignore the constants and the plus and minus signs, which just represent direction, in the revised version of Maxwell’s equations, the first equation is exactly the same as the second equation, except with the E and B (and little e and b subscripts) interchanged. The same is true of the third and fourth equations: they are exactly the same as each other except the places of E and B, and e and b, are switched. This is the symmetry physics are looking for when they search for magnetic monopoles.

Until next time….

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5 Responses to “More Magnetic Monopoles, With a Summery Hint of Maxwell’s Equations”

  1. Jenn Says:

    That makes a lot of sense! Perhaps I am turning into a physicist like my labmates believe I am, as I find the idea that these things round out Maxwell’s equations possibly more convincing than I ought to in wondering whether magnetic monopoles might exist. 🙂 Thanks for breaking it down.

  2. Brian Says:

    I don’t understand like any of this once you got to the math part. I just couldn’t seem to get it. But I think it’s safe to say it’s less your fault and more that I just don’t like witchcraft.

  3. dyslexicmathematician Says:

    To Brian:

    Yeah, I figured I was likely to lose a lot of people with all that. This post was originally a response to Jenn from the last post, and she’s a chemistry grad student, so I knew she could follow the math. I went into the math with people like her in mind, but I recognize it’s very dense and difficult, if not impossible, for those with limited higher math background to follow from just what I’ve said here. It’s not your fault; there’s always something that could have been explained differently, or in this case, probably more slowly with more examples.

    Hopefully the beginning and end made sense to everyone though. The middle was extra credit.

  4. spiroz Says:

    I read today that the government is trying to drop the “doomsday” lawsuit against against Cern’s LHC. What is your prediction if a black hole or rogue monopole is created? Will we be sucked or blown into oblivion?

  5. dyslexicmathematician Says:

    To Spiroz:

    This is admittedly beyond my area of expertise, but I’ve read that mini black holes, if they do exist, should have a near infinitesimally small lifespan such that they are harmless. If black holes can be created by the LHC, then they should be created every now and then by high-energy particles colliding in the upper atmosphere too, and we remain unharmed by this. This goes for the worries about all the weird forms of matter the LHC could create, actually: collisions at these energies already occur in our atmosphere, so if these forms of matter can exist, they have existed in our atmosphere throughout the Earth’s history and do not appear to pose any threat.

    I realized that this answer to your question was probably somewhat unsatisfying, so I did some poking around on the internet for more info. This is the summary from the report done recently by the LHC Safety Assessment Group, available on CERN’s website:

    The safety of collisions at the Large Hadron Collider (LHC) was studied in 2003 by the LHC Safety Study Group, who concluded that they presented no danger. Here we review their 2003 analysis in light of additional experimental results and theoretical understanding, which enable us to confirm, update and extend the conclusions of the LHC Safety Study Group. The LHC reproduces in the laboratory, under controlled conditions, collisions at centre-of-mass energies less than those reached in the atmosphere by some of the cosmic rays that have been bombarding the Earth for billions of years. We recall the rates for the collisions of cosmic rays with the Earth, Sun, neutron stars, white dwarfs and other astronomical bodies at energies higher than the LHC. The stability of astronomical bodies indicates that such collisions cannot be dangerous. Specifically, we study the possible production at the LHC of hypothetical objects such as vacuum bubbles, magnetic monopoles, microscopic black holes and strangelets, and find no associated risks. Any microscopic black holes produced at the LHC are expected to decay by Hawking radiation before they reach the detector walls. If some microscopic black holes were stable, those produced by cosmic rays would be stopped inside the Earth or other astronomical bodies. The stability of astronomical bodies constrains strongly the possible rate of accretion by any such microscopic black holes, so that they present no conceivable danger. In the case of strangelets, the good agreement of measurements of particle production at RHIC with simple thermodynamic models constrains severely the production of strangelets in heavy-ion collisions at the LHC, which also present no danger.

    I think the smart money says that nothing cataclysmic will happen due to any of these objects, if they are created. The report is illuminating. Though it’s not entirely accessible to the lay reader in its word choice, it’s not entirely unaccessible either. I recommend reading it, if you’re interested.

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