Revisiting Magnetic Monopoles

In this essay I want to revisit the topic of magnetic monopoles, the magnetic analogue of electric charge, and re-examine how they fit into Maxwell’s Equations and whether they conform to the conservation laws. Let’s start with the standard version of Maxwell’s Equations, the Lorentz force law, and the Lorenz condition:

(Eq’ns 1)

Those equations properly apply only to electrically charged particles in vacuum, but we can use them in our imaginary experiments by asserting that any non-particulate matter display the electromagnetic properties of vacuum.

Gauss’s law of the electric field, the first of those equations, tells us that if we have a positive electric charge Q at some point in space, then the electric field vectors emanating from that charge, in order to make the divergence positive, point away from the charge. In accordance with that statement we define a positive magnetic charge M such that its density σ conforms to the divergence of the magnetic field in accordance with

(Eq’n 2)

with the magnetic field lines pointing away from the positive charge. Experiments conducted with electromagnets and permanent magnets determine that a positive magnetic charge corresponds to a north magnetic pole. Because an electric charge generates an electrostatic field in accordance with the negative gradient of an electrostatic potential φ, we can also say that a magnetic charge generates a magnetostatic field in accordance with the negative gradient of a magnetostatic potential ψ.

Without its time-derivative term, the fourth of Equations 1 tells us that an electric current generates a magnetic field and does so in accordance with the right-hand rule: if I grasp a long, straight wire with my right hand in such a way that my thumb points parallel to the flow of the electric current (the direction in which positive charge moves), then my fingers curl in the direction in which the associated magnetic field circulates; in other words, if I look along the wire with the electric current moving away from me, the magnetic field lines appear to me to circulate clockwise about the wire.

Assume that an electric current moves in a long, straight wire in the positive direction along the x-axis in the standard Cartesian coordinate frame. A short distance in the positive y-direction from that wire we establish a long, straight wire parallel to the electric wire and establish a stationary positive magnetic charge on it. In that situation the magnetic field circulating around the electric wire points in the positive z-direction where it crosses the magnetic wire and thus it exerts a positive z-ward force upon the magnetic charges. The magnetic field pointing directly away from the magnetic wire points in the negative y-direction where it crosses the electric wire, so the motion of the electric current across it produces a force that pushes the electric wire in the negative z-direction.

If an observer moves with the electric current, occupying an inertial frame in which the positive charges making up the current stand motionless, then that observer sees the magnetic charges moving in a current in the negative x-direction. For that observer the forces must act in the same directions as they do for us and they must have magnitudes that equal each other (lest the system violate the law of conservation of linear momentum). In order that the force on the electric wire point in the negative z-direction the magnetic current must generate an electric field that circulates around the magnetic wire in such a way that it points in the negative z-direction where it touches the electric wire: if we look in the direction in which the magnetic current flows we see the electric field lines circulating counterclockwise around the magnetic wire. Because that electric field must have a non-zero curl where it crosses the magnetic wire, we must modify Faraday’s law to read

(Eq’n 3)

in which **k** represents the magnetic current density, essentially equal
to the product of the magnetic charge density in the wire and the velocity at
which the charges move (assuming, for convenience, that they all move at the
same speed). We can prove and verify that equation by calculating its divergence
and substituting from Equation 2 for **B**: we get

(Eq’n 4)

the equation of continuity for magnetic charge. If magnetic charge obeys a conservation law (and we expect that it does), then those two equations stand true to Reality insofar as magnetic charge actually exists.

Also for that observer the electric charge must exert a force upon the magnetic current, a force that points in the positive z-direction. Again we have a left-hand rule rather than a right-hand rule giving us the direction in which the force acts in accordance with the vector cross product of the velocity of the current upon the field. In order to work out the actual mathematical form of that force we must first take a slight detour into a discussion of the electrotonic and magnetotonic fields.

We have already deduced the existence of the electrotonic
field (represented mathematically by the magnetic vector potential **A**) as
a relativistic extension of the electrostatic potential. If we have electric
charges producing a potential φ
at a point and if all of those charges move at the same velocity **v**, then
they also generate at that point an electrotonic field

(Eq’n 5)

The full description of the electric field then becomes

(Eq’n 6)

The electrotonic field also generates a magnetic induction field through its curl,

(Eq’n 7)

Given the existence of magnetic charge, we expect that a set of charges that
produces a magnetostatic potential
ψ at a point and that all move with
the same velocity v will also generate at that point a magnetotonic field,
represented mathematically by the electric vector potential **T** through the
equation

(Eq’n 8)

We assume, by analogy with the electrotonic field, that the magnetotonic field participates in the basic force equation in the form

(Eq’n 9)

and that its curl describes the generation of an electric field,

(Eq’n 10)

To test that assumption we take the curl of the negatives of Equations 6 and 9 to get

(Eq’n 11)

and

(Eq’n 12)

Using the last terms in those two equations to make the appropriate substitutions from Faraday’s law and the current-free version of Ampere’s law, we get those equations as

(Eq’n 13)

and

(Eq’n 14)

Undoing the curl in Equation 14 gives us

(Eq’n 15)

which conflicts with Equation 9, both in the algebraic sign and in the
coefficient on **B**. We can correct the problem with the algebraic sign
easily enough by simply reversing the algebraic sign on **T** in Equation 10.
That move makes sense because the magnetic current generates an electric field
that conforms to a left-hand rule, the negative of a right-hand rule.

The coefficient of **B** in Equation 15 gives a harder
problem. We know that in vacuum **B**=μ_{0}**H**,
so we need only replace **B** in Equation 9 with **H**, thereby absorbing
the magnetic permeability of vacuum into **T**. We can also absorb the
electric permittivity of vacuum into T be rewriting Equation 10 as

(Eq’n 16)

and Equation 7 as

(Eq’n 17)

Then we must rewrite Equation 9 as

(Eq’n 18)

In this way we redefine the magnetostatic potential and the magnetotonic field to conform to the requirements of mathematical consistency.

So now we can rewrite Equations 1 in a nicely symmetric form:

Eq’ns 19)

We can see in those equations the ghost of an electromagnetism in which both electric charge and magnetic monopoles generate electric and magnetic fields and interact with them. Mere mathematical description seems to prescribe the actions of the fundamental components of Reality: it certainly conforms to what we measure insofar as we can measure the relevant phenomena. We have come to expect that conformity of anything that we put on the Map of Physics.

But now the system displays an effect that we should notice and examine further. The force exerted on the electric wire pushes the wire downward and the force exerted on the magnetic wire pushes that wire upward, so the foundation holding the wires receives a torque oriented, in accordance with the right-hand rule, in the positive x-direction. This torque appears to violate the conservation law pertaining to angular momentum because we can’t see a source for the counter-torque that would bring the system into compliance with the rotary version of Newton’s third law of motion. However, we have ignored the negative electric and magnetic charges necessitated by the conservation laws that those entities must obey. We need to see whether those charges introduce into the system a torque that cancels the one that we have already discerned.

Let’s reconceive our imaginary experiment. First we
establish a source at the point (-x_{0},0), a sink at the point (+x_{0},0)
and a station at the point (0,0), the origin of the coordinate grid. Then we
carry out a six-step process that moves point-like manifestations of electric
and magnetic charges through that system:

1. At the source we separate a neutral body into two minuscule bodies that carry equal amounts of positive and negative magnetic charge (+M and -M) and we move +M to the station at the point (0,0);

2. At the source we separate a neutral body into two
minuscule bodies that carry equal amounts of positive and negative electric
charge (+Q and -Q), then we move +Q along a straight line to the point (0,+y_{0})
and thence move it straight to the sink;

3. Move +M to the sink;

4. Move -M to the station at the point (0,0);

5. Move -Q straight to the point (0,+y_{0}) and
thence move it to the sink, where it neutralizes +Q;

6. Move -M to the sink, where it neutralizes +M.

In steps 1, 3, 4, and 6 that apparatus generates no torque, because the magnetic charges move straight along electric field lines and exert no forces upon the electric charges. In each of steps 2 and 5 we get two contributions to the torque that the apparatus generates – a positive one from the interaction between the electric charge and the magnetic charge at the origin and a negative one from the interaction between the electric charge and the magnetic charge at either the source or the sink. Normally I would now calculate the torques, devising an algebraic description of them in terms of the parameters of the apparatus and the procedure by which it operates, but in this case I have three reasons for not doing so:

a. the calculations involve two hideously ugly integrals whose solutions would take more time than I want to invest, because

b. I can use a hand-waving, relative-magnitude argument to prove and verify the proposition that the imaginary experiment creates angular momentum out of nothing in blatant violation of the conservation law pertaining to angular momentum, and

c. we don’t need to calculate a detailed description of forces and torques for any phenomenon that will never, under any circumstances, occur.

The hand-waving argument goes like this:

We assume, for convenience, that we have made the apparatus with y0 much smaller than x0. In step 2 we have +Q crossing the magnetic field lines emanating from +M. The force that +Q receives from +M and the force that it exerts on +M through the magnetic field that its motion generates produce a positive torque (which we define in this case as a torque whose vector points in the positive x-direction). Also in step 2 +Q has no interaction with -M during the first half of its traverse because it moves directly along one of -M’s magnetic field lines. During the second half of the traverse +Q crosses -M’s magnetic field lines and thereby, along with the force that it exerts upon -M, generates a negative torque. At each point on the second half of the traverse the force exerted upon +Q by +M exceeds the force exerted upon +Q by -M (and vice versa through Newton’s third law), which we know because 1) the electric charge comes closer to +M than to -M (making the magnitude of the field emanating from +M larger for +Q than that of the field emanating from -M) and 2) the angle between +Q’s velocity and the magnetic field lines is greater for the lines emanating from +M than for those emanating from -M (the cross product describing the force stands in direct proportion to the sine of that angle); thus, the contribution to the system’s torque from any part of the second half of +Q’s traverse is guaranteed to be larger from +M than from -M, so the integration over that part of the traverse yields a net positive angular momentum. The same analysis applies to step 5, the only difference being that the negative contribution to the net torque comes from the first half of -Q’s traverse instead of from the second half. In both steps the positive contribution outweighs the negative contribution, so the system creates a net positive angular momentum while returning to its original neutral state, in violation of the conservation law pertaining to angular momentum. Q.E.D.

We have already inferred the law of conservation of angular momentum from a fundamental symmetry of space, so we won’t give it up easily. Instead we dismiss the possibility that magnetic monopoles exist. But in doing that we re-introduce asymmetry back into Maxwell’s Equations (we must use Equations 1 rather than Equations 19). We may take that fact as a hint that asymmetry bears somewhat the same relation to matter and forcefields that symmetry bears to space and time. That’s a topic we need to explore later.

habg