The Question of Magnetic Monopoles

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    In 1931 Paul A.M. Dirac hypothesized the existence of a particle carrying the magnetic analogue of the electric charge that exists on electrons. He drew on quantum-theoretical arguments and described a particle surrounded by a magnetic field whose ętherial lines of force, rather than form the usual closed loop, would radiate straight away from the particle over indefinite distances. Because physicists, in order to calculate magnetic fields, had long employed the useful fiction that the pole faces of magnets were covered with such particles, we call those particles magnetic monopoles. Now, if only we could find some....

    So far they have eluded all attempts to find them. Every now and then some physicist claims to have found evidence of their presence in some experiment, but follow-up research has always come up empty. When all of our attempts to find something fail, we can infer two reasons for the failure: either we are not looking in the right place (unlikely in this case) or the thing does not exist. In what follows I will show you strong reasons to accept the second of those options as the correct explanation for physicists' failure to find Dirac's monopoles.

    Let's start with the basics. If we draw Maxwell's Equations as they apply to a region of space in which we find no electric charge or electric current, then we see a remarkable symmetry between the electric and magnetic fields reflected in those equations;

(Eq'ns 1, 2, 3, & 4)

From an examination of those equations we would be unable to determine which of the fields, electric E or magnetic H, we should accept as the more fundamental.

    That fact brings to mind Einstein's introductory comments in his first paper on Relativity, "On the Electrodynamics of Moving Bodies". Einstein described a situation in which he envisioned a conductor (presumably a wire) and a magnet with some relative velocity between them. He then went on to describe the situation as observers at rest relative to the conductor and to the magnet, respectively, would measure an electric field driving a current in the conductor and infer an electromotive force generated by the conductor plowing through the magnetic field. From the equivalence of the two points of view Einstein went on to discuss the theory of Special Relativity. But the hint at a symmetry between the electric and magnetic fields remains.

    If we put electric charges and currents into the region under study, we restore to Maxwell's Equations the asymmetry that leads us to regard the electric field as more fundamental than the magnetic field. But what could we say if we discovered the existence of magnetic charge, magnetic monopoles that emit radially symmetric magnetic fields in the same manner in which electric monopoles, such as those manifest in the electron, emit radially symmetric electric fields? Primarily, we can say that the existence of magnetic charge would restore to Maxwell's Equations the symmetry that we find in the vacuum form of those equations. If we represent magnetic charge density with λ and magnetic current density with h, then Maxwell's Equations would take the form

(Eq'ns 5, 6, 7, & 8)

So does magnetic charge exist?

    If we represent the amount of magnetic charge with the letter p (just as we represent the amount of electric charge with the letter q), then we must have the analogue of Coulomb's law of electric force telling us that between two particles carrying magnetic charges p1 and p2 and located a distance r from each other in vacuum we must measure a force

(Eq'n 9)

in which µ0 = 4x10-7 webers per ampere-meter, the magnetic permeability of vacuum. In accordance with established convention, we specify that magnetic charge comes in north poles (positive magnetic charge) and south poles (negative magnetic charges) and that the magnetic induction field B points from north poles to south poles. The magnetic induction field that a given magnetic pole p generates at a distance r from the pole then equals

(Eq'n 10)

in which the magnetostatic potential

(Eq'n 11)

From that equation we obtain the electric vector potential

(Eq'n 12)

such that

(Eq'ns 13 & 14)

which leads us to the magnetoelectric analogue of the Lorentz force, the force that an electric field E exerts upon a body carrying a magnetic charge p and moving at velocity v,

(Eq'n 15)

    Now imagine that we have two small particles connected to each other with a thread and that we have positioned the particles on our x-axis. Into the particle occupying the point (-x0, 0, 0) we infuse a positive electric charge q and into the particle occupying the point (+x0, 0, 0) we infuse a north magnetic charge p. And now we can watch those two particles do absolutely nothing. We know that we will see nothing happen because we know that the magnetic field emanating from the magnetic charge exerts no force upon the electric charge and that the electric field emanating from the electric charge exerts no force upon the magnetic charge.

    An observer moving in the positive y-direction at some speed relative to us will disagree with that assessment. In that observer's frame our two particles have a velocity in the negative y-direction, so each of the particles will manifest the appropriate Lorentz force due to its interaction with the other particle's field. Applying the right-hand rule, we can see that for that observer the moving electric charge will interact with the magnetic field (pointing initially in the negative x-direction) and take from it a force pointing in the negative z-direction and the moving magnetic charge will interact with the electric field (pointing initially in the positive x-direction) and take from it a force pointing in the positive z-direction; thus, as the observer looks in the positive y-direction they see this thing we have set up begin to turn in the counter-clockwise sense.

    As the device spins progressively faster the motions of the particles in the x-z plane generates a force in the positive y-direction in each of the particles, so the device accelerates in that direction. As it does so, as it comes closer to matching speeds with the observer, the torque that it exerts upon itself diminishes, going to zero when the device has zero translational speed relative to the observer. Still spinning, it continues to accelerate, moving ever faster in the positive y-direction relative to the observer, which velocity causes the device to exert a clockwise torque upon itself. That torque slows the device's rotation and, thus, its acceleration, until the device stops turning altogether. But because the device has a velocity relative to the observer, the torque continues to operate, making the device spin clockwise. Spinning clockwise, the device exerts upon itself a force that drives it in the negative y-direction, slowing it to a stop and then giving it a negative y-velocity. Again the torque reverses and the device stops spinning just as it achieves its original speed in the negative y-direction. The device then repeats that cycle for our observer endlessly, acting in a manner analogous to that of the Wilberforce pendulum.

    So our simple device immediately violates the law of noncontradiction. In our frame the device displays no acceleration, either linear or rotary. But in the frame of an observer moving past us at a uniform speed it displays both reciprocating linear and rotary accelerations. We have no way in which we can reconcile those two sets of observations through any simple transformation based on unaccelerated relative velocity (like the Lorentz Transformation), so those observations, however imaginary, represent a contradiction.

    But we can do something else illegitimate with our magnetic charge. Imagine that we have a vehicle that resembles a twin-engine propeller-driven airplane. In place of the propellers we mount batons on the axles of the motors and affix knobs at their ends. On the starboard baton we put a north magnetic charge into one knob and a positive electric charge into the other and on the port baton we put a north magnetic charge into one knob and a negative electric charge into the other. When the motors counter-rotate the batons, the forces that the batons exert upon themselves pull the craft forward and accelerate it. As long as the batons spin, the craft accelerates in blatant violation of the law pertaining to conservation of linear momentum.

    Now we have a simple syllogism: 1) anything that enables us to violate the law of noncontradiction or an absolute conservation law cannot exist in Reality; 2) magnetic monopoles enable us to violate the law of noncontradiction or an absolute conservation law; therefore, 3) magnetic monopoles cannot exist in Reality.

    Of course, you know what this means. When we get into the quantum theory we will have to revisit this topic and Dirac's deduction of the existence of magnetic monopoles to see what we can learn.

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