On Electromagnetic Angular Momentum and the Magnetic Field
Fig. 1 depicts an uncharged, disc-shaped ceramic magnet of thickness D, radius R, and central hole radius H. Note the polarity: the magnetic field lines emerge near the periphery and enter near the central hole. The magnet is mounted on a non-ferrous drive shaft and can be spun around the y-axis.
Disc-Shaped Magnet in Cross Section
Since the charge density is zero everywhere, there is no E field. The magnetís B field can be modeled as the sum of the B fields of an array of microscopic, uncharged current loops. For the polarity shown, each tiny current loopís plane is normal to a radial line from the magnetís center to its periphery. Fig. 2 depicts one such loop whose center lies beneath the positive x-axis. It is assumed that the negative charge is at rest and the positive charge circulates counterclockwise.
Microscopic Current Loop
If the magnet in Fig. 1 spins with w pointing in the positive y-direction, then the instantaneous velocity vector of each microscopic current loopís center lies in that loopís plane. In such cases application of the Lorentz transformation indicates that the charge density at any moment is not zero at all points around the loop. For example, if the loop depicted in Fig. 2 moves in the negative z-direction, then the upper leg will be negatively charged and the bottom leg will be positively charged. The greater the translational speed, the greater the electric polarization.
Collectively such polarizations result in a positive charge on the magnetís lower surface and a negative charge on its top surface. In effect the spinning magnet is also a parallel plate capacitor. But, since v = wr, the capacitorís surface charge density increases with distance from the center.
If the magnetís thickness is sufficiently small, then the external E field will be practically zero. Within the magnet, however, E(r) will be in accordance with Gauss and will nominally point in the positive y-direction. B points radially outward at all internal points. And since div B = 0, dB/dr is zero and |B| is single-valued inside the magnet. It follows that the electromagnetic momentum density, g = eoE X B, points tangentially, in the same direction as the local velocity, and has a magnitude that increases with distance from the center.
By symmetry any increment of translational momentum on one side of the magnet is canceled by a diametrically opposed increment. The total translational momentum is thus zero for all w. But the increments of angular momentum add, and the spinning magnet has a net electromagnetic angular momentum, in addition to its mechanical angular momentum. The electromagnetic angular momentum (and hence the total angular momentum) grows as w increases.
The volume of a ring at distance r from the center is
Thus the electromagnetic angular momentum of that ring has a magnitude of
The total electromagnetic angular momentum magnitude is thus
Since the direction of this electromagnetic angular momentum is the same as the discís mechanical angular momentum, whenever w > 0 a greater torque must be applied to attain a given angular acceleration of the magnetized disc, than would be needed in the case of an unmagnetized disc of equal mass. The same would be true if (a) the magnetís polarity is reversed, (b) w is reversed, or (c) both the polarity and w are reversed.
Fig. 3 depicts a magnetic polarity discussed in other articles on this site.
Alternate Polarity, Disc-Shaped Magnet
For the polarity shown in Fig. 3, the plane of each microscopic, uncharged current loop would be normal to a vertical line through its center. With w = 0 there would again be no mechanical angular momentum. And, although each current loopís angular momentum points in the positive y-direction, to the extent the microscopic loop currents are constant there would be no measurable electromagnetic angular momentum. But if it were possible to drive each microscopic loop current to zero (by demagnetizing the disc), then the net electromagnetic angular momentum of all the microscopic current loops would be manifest as a final, mechanical angular momentum of the disc. Perhaps the moral of the present article is that this will not always be the case. In Fig. 1 the angular momenta of the microscopic current loops cancel in pairs. They do not add, as they do in Fig. 3. In the Fig. 1 case, then, there would be no final mechanical angular momentum if the disc were to be demagnetized!