Magnetic Monopoles Revisited
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With regard to electric and magnetic fields, we know that one emanates from point-like sources, monopole charges, and the other doesn’t. Electric monopoles, usually manifested in electrons and atomic nuclei, are common and easily produced for experiments: no solid evidence has ever been found for the existence of magnetic monopoles. We can say with some confidence that magnetic monopoles do not – indeed, cannot – exist. That confidence comes from our understanding that, if magnetic monopoles existed, we could use them to contrive blatant violations of some of the most fundamental laws of physics.
Imagine a Cartesian coordinate grid floating in space. Mark two points on the x-axis at x=+R (to the right of the origin) and at x=-R (to the left of the origin) and extend straight lines through those points parallel to the z-axis. Centered on each of the two points construct circles of radius r0<<R and extend through each point on those circles a straight line parallel to the z-axis. On the tubes thus defined lay long, straight wires that nest together, alternating electrical conductors with electrical insulators (such as copper and glass). This construction ensures that electric current will flow only along the length of each tube and not around its circumference.
On a thread lying on the centerline of the left-hand tube, the line that passes through x=-R, we spread a uniform distribution of positive magnetic poles (poles with the B-field pointing away from them), so many poles per meter. On a thread lying on the centerline of the right-hand tube, the line that passes through x=+R, we spread a uniform distribution of negative magnetic poles, the magnitude so chosen that the net magnetic charge on the two threads equals zero.
The system also carries no net electric charge. I assert that at a great distance from the origin of our coordinate frame, in the positive and negative z-directions, cables connect the two tubes together, thereby creating a closed electrical circuit. A battery or a generator attached to one of those cables provides a direct current voltage that drives and electric current of magnitudeI around the circuit. In the left-hand tube electrons flow in the negative z-direction (away from us), so the current flows in the positive z-direction (toward us). In the right-hand tube the current flows in the negative z-direction.
No electric fields exist in or around this system. Magnetic fields, due to the poles on the threads and to the electric current in the tubes, exist throughout the system. The forces exerted within the system are thus manifestations of magnetic interactions only. The fields emanating from the threads conform to straight lines radiating from the threads in the x- and y-directions only, the field lines pointing away from the left-hand thread and pointing toward the right-hand thread. Each of the tubes with its electric current produces a magnetic field only outside itself, the lines of force conforming to circles centered on the thread inside the tube. The field around the left-hand tube turns counterclockwise and the field around the right-hand tube turns clockwise.
The magnetic field emanating from the left-hand thread has strength B0 where it crosses the left-hand tube and strength rB0/2R where it crosses the right-hand thread. We can make that same statement about the field emanating from the right-hand thread by simply exchanging left and right in that first statement. Through their fields the two threads exert upon each other forces that are equal and oppositely directed along the x-axis; thus, the threads, vis-a-vis each other, exert no net force and no torque upon the system.
The electric current flowing in the left-hand tube interacts with the left-hand thread’s field and produces a torqueIB0r turning counterclockwise. The current flowing in the right-hand tube exerts a force IB=IB0r/2R in the negative y-direction (down) and, thus, a torque IB0r turning clockwise. So the two torques counterbalance each other, but we have a net force IB0r/2R exerted in the negative y-direction. The same analysis applies to the electric current in the tubes acting on the magnetic field emanating from the right-hand thread. In that case we also get no net torque, but we have a net force IB0r/2R pushing the left-hand tube in the positive y-direction (up). We thus get no net force acting on the system, but we get a net torque of IB0r turning the system clockwise about its central axis (the z-axis of our coordinate grid).
The magnetic field generated by the current flowing in the left-hand tube points in the positive y-direction (up) where it crosses the right-hand tube. The current flowing in the right-hand tube thus exerts a force in the positive x-direction (to the right). The magnetic poles on the right-hand thread exert a force in the negative y-direction, which corresponds to a clockwise torque about the system’s central axis. The magnetic field generated by the current flowing in the right-hand tube also points in the positive y-direction where it crosses the left-hand tube. The current flowing in the left-hand tube thus exerts a force in the negative x-direction, thereby counterbalancing the force exerted on the right-hand tube. The magnetic poles on the left-hand thread exert a force in the positive y-direction, which corresponds to a clockwise torque about the system’s central axis. So the interaction between the poles and the electric current produces a net clockwise torque about the system’s z-axis.
If we add up all of the interactions within the system, we get no net force (in conformity with the conservation of linear momentum theorem) and a net clockwise torque (in conflict with the conservation of angular momentum theorem). We have deduced the theorem pertaining to conservation of angular momentum from the basic facts about the existence of the Universe, so we need strongly compelling reason to abandon that theorem or to modify it. Therefore, we infer that either electric or magnetic monopoles cannot exist. We know that electric monopoles exist, so we infer that magnetic monopoles cannot – and do not – exist. Q.E.D.
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