On The Physics of Homopolar Generators and Motors
1. A Homopolar Generator.
Fig. 1_1 depicts a cross-sectional view of a disc-shaped permanent magnet, above which a conducting probe can be rotated counterclockwise (when viewed from positive y). When the probe is spun it is assumed to slide frictionlessly along a conducting ring that is glued to the magnet. A closing wire, with one end soldered to the ring and the other sliding on the conducting probe drive shaft, completes the circuit.
The apparatus constitutes a homopolar generator. When the probe cuts across the discís magnetic field lines, the conduction electrons in the probe experience the Lorentz magnetic force qvXB radially inward along the probe. The resulting electron flow comprises a conventional current outward along the probe. The faster the probe is spun, the greater the current.
A Homopolar Generator
2. Driving the Generator.
Since magnetic forces do no work, a question arises concerning how the generator current can expend power on the load. A no doubt simplistic model for the creation of a working electric current in the probe is as follows. When the probe drive shaft is rotated (by some driving agent), conduction electrons at a given distance from the shaft experience a radial magnetic force toward the drive shaft. This force is always at right angles to their velocity and does not change their speed. But it does decrease their tangential component of velocity. Consequently the "back" side of the probe overtakes and elastically collides with them, increasing their energy by increasing their tangential velocity. The process repeats until the energized electrons exit the probe and enter the (conducting) drive shaft. They continue on, through the load, where their newfound energy is spent.
It is noteworthy that this model necessarily includes a non-electromagnetic phenomenon, namely the collision mechanism between electrons and the probeís overtaking backside. Each such collision results in a clockwise increment of torque on the probe. In order to maintain a given probe spin rate, the driving agent must apply a counterclockwise torque to the probe drive shaft.
3. The Torque on the Disc-shaped Magnet.
The permanent magnet can be conveniently modeled as an array of microscopic, uncharged, square current loops. Viewed from positive y, current circulates counterclockwise around each such loop. To the extent the tiny loops abut, Stokesí theorem suggests that the entire array amounts to a current circulating counterclockwise around the disc-shaped magnetís periphery. We shall refer to this current as the "megaloop".
The magnetic field of the current-conducting probe (momentarily situated slightly above the right half of the megaloop in Fig. 3_1) has components that point into the page on the megaloopís bottom (z>0) right half, and out of the page on its top (z<0) right half. Given the counterclockwise current circulation around the megaloop, the right half experiences a net force toward negative z. There is a counterclockwise torque on the disc-shaped magnet.
Megaloop and Probeís B-Field Components
If the magnet is not allowed to spin, an external agent must counteract this torque with a clockwise torque. For example, the disc-shaped magnet might be mounted on its own shaft, with the external agent applying the necessary clockwise torque to the shaft. Since the magnet does not spin, no power is expended in applying this torque.
4. The Spinning Magnet and Resting Probe Alternative.
Let us now reverse the situation and consider the case where the probe (Fig. 1_1) is at rest and the disc-shaped magnet, closing wire, etc. rotates clockwise. Up until the turn of the century conventional wisdom supposed that in this case there would be no current induced in the probe circuit. (The leg of the closing wire that is parallel to the magnetís surface is presumably far enough removed that any magnetic forces along it are practically negligible.) The rationale was that, as in the first case, the (uncharged) spinning magnetís B field does not vary in time. Consequently there is again presumably no electric field associated with the magnet. And the conduction electrons in the probe, now nominally at rest, experience no magnetic force.
It turns out that conventional wisdom was wrong! Measurements made by Guala-Valverde and others demonstrated that in this case too a current is induced in the probe circuit. The question is, what physically explains the measured current?
The answer lies in an application of the Lorentz transformation to an uncharged current loop. Let us say the loopís rest frame is that frame in which the loopís center is at rest. In the rest frame, charge of one sign is at rest and charge of the opposite sign circulates at a constant speed around the loop. Both charges are evenly distributed around the loop.
Viewed from a frame in which the loop translates along a line lying in its plane, the circulating charge at any instant is not evenly distributed around the loop. There is a preponderance of circulating charge in one leg (again assuming a square loop, appropriately oriented) and a surfeit in the opposing leg. Consequently the moving loop has a nonzero electric dipole moment whereas the resting one has none. The moving loop accordingly has a dipolar electric field (whereas the resting one has no electric field).
For clockwise rotation of the disc-shaped magnet, and counterclockwise circulation of each microscopic loopís current, each microscopic current loop will have a dipole moment that points radially inward. Collectively these tiny electric dipoles engender a conservative electric field with a component that points radially outward at any point above or below the spinning, disc-shaped magnet. This resultant electric field induces probe conduction electrons to flow inward toward the (resting) probe shaft, quite as they did in the first case where the magnet was at rest and the probe was rotated.
Since the rotating magnetís electric field is conservative, it might be wondered why the emf around the entire probe/closing wire circuit is not zero. The answer is found in the fact that the total force on a test charge that is at rest relative to the magnet is zero. The magnetic force on such a test charge is equal but oppositely directed to the electric force. This is not the case for charges in the probe. Being nominally at rest, these charges experience only the spinning magnetís electric field, and consequently there is a net emf around the entire circuit.
As in the first case, external agents must exert a counterclockwise torque on the (now resting) probe shaft. And a clockwise torque must again be applied to the magnet shaft. But the emf in the probe circuit is now a function of the magnetís spin rate. (The greater the magnetís spin rate, the greater the electric field driving electrons through the probe.) Thus the power dissipated by the load is now a function of the power expended to spin the magnet. In this case it is of course the torque required to hold the probe at rest that expends no power.
But why should the resting probe experience a torque in this case? The spinning magnetís radial electric field drives conduction electrons along the probe and not at right angles to it. The torque on the probe results from the magnetic forces experienced by the electrons as they drift along the probe. These forces cause the electrons to accelerate at right angles to their velocity, periodically colliding with the (resting) probeís bottom wall and collectively resulting in a clockwise torque on the probe. So far as the net torque is concerned, matters are quite as they were when the wall collided with the electrons in the first case.
5. Homopolar Motors.
Conventional electric generators can also serve as electric motors. This is also true for the homopolar generators discussed above. For example, if the load in Fig. 1_1 is replaced with a battery that drives probe electrons radially inward toward the probe shaft, then the probe will rotate clockwise (not counterclockwise, as it does in the generator case) and be capable of doing mechanical work in the process. (i.e. the agent that drives the probe shaft, in the generator case, is replaced with a mechanical load.)
Thanks to the reciprocity discovered by Guala-Valverde et al, the second (spinning magnet/stationary probe) configuration can also serve as a motor, with the magnetís shaft now providing mechanical power. The force acting on the probe-half of the megaloop at any given instant again constitutes a counterclockwise torque on the magnet and its shaft. This torque will cause the magnet to spin counterclockwise (not clockwise), and the spinning magnetís shaft can be made to do mechanical work.
It is noteworthy that, when the disc-shaped magnet rotates counterclockwise, its electric field above and below the magnet points toward the axis of rotation. The resulting emf in the probe thus runs counter to the radially outward probe current generated by the battery. The battery must counteract this emf in order for the current to flow radially outward in the probe. Since there is now no load in the probe circuit, the resistance around it is practically zero. All of the power output by the battery to overcome the magnetís counter-emf thus equates to the product of the torque on the magnet times its angular rate of rotation. That is, the mechanical power delivered by the magnet shaft equates to the power output by the battery.