On the Electromagnetic Inertia of Current Loops
Fig. 1 shows a non-accelerating, square, positive line charge circulating CCW on an equal, non-circulating, negative line charge. l, the total line charge density, is constantly zero at all points and hence E=0 everywhere. There is a dipolar B field.
An Uncharged Current Loop
It is readily shown that such a loop is electrically polarized when it translates in its plane. For example, if the loop is viewed from inertial frame K’, which moves in the positive x direction, then from that perspective the net line charge density in the top leg will be positive, and that in the bottom leg will be negative. At the loop’s center there will be a nonzero E’ field pointing in the –y’ direction.
If the loop is accelerating in the negative x’ direction, then the positive line charge density in the top leg will be increasing with time. Similarly for the negative line charge density in the bottom leg. Consequently dEy’/dt’ at the center point will be negative. This will induce a component of circulating B’ that points toward negative z’ in the left leg, and toward positive z’ in the right leg. The current in the left leg will accordingly experience a magnetic force toward +x’. And the current in the right leg will also experience a magnetic force toward +x’. In general the loop will react to any external agent that attempts to accelerate it in its plane; the loop will have inertia.
Since the reaction force is magnetic, we can say that the loop has electromagnetic inertia quite apart from (and in addition to) any mechanical inertia. Whereas the electromagnetic inertia of a particle of charge can be attributed to an acceleration-induced electric field right at the charge, however, the inertia of the current loop can be attributed to an acceleration-induced magnetic field on the "leading" and "trailing" legs.
In a previous article it was discussed how a permanent, ceramic, disc-shaped magnet will electromagnetically resist having its angular momentum changed. To the extent the magnet can be modeled as an array of microscopic, uncharged current loops, the foregoing discussion sheds light on what the seat of this angular inertia is.
Like its omni-present B field, a moving loop’s electric field has energy. If EB is the magnetic field energy, and EE is the electric field energy, then the loop’s electromagnetic mass is
When the loop’s center is at rest, then EE=0 and
More generally we expect that
The force needed to accelerate the loop is theoretically
It is noteworthy that a loop need not be uncharged in order to develop a dipolar E field when it translates. For example, if the loop in Fig. 1 were purely positive charge, then the positive charge density in the top leg would be greater than that in the bottom leg when the loop translates to the left. dE/dt would still be nonzero at the center when the loop accelerates, and the left and right legs would still experience a reactive magnetic force. Of course the charge increments around the loop would also experience acceleration-induced electric forces. Evidently the total electromagnetic mass of a particle like an electron (with net charge and nonzero magnetic dipole moment) can be attributed to a combination of electric and magnetic reaction forces.