Fig. 3.1_1 depicts a positive test charge, at rest in inertial frame K, above an infinitely long, uncharged line current. The current consists of equal-density, opposite-sign line charges with one sign moving to the left and the other to the right.

Having a net charge density of zero, the line current engenders no electric field. The test charge accordingly experiences no electric force. The line current does have a magnetic field. But, being at rest, the test charge also experiences zero magnetic force. The total Lorentz force on the test charge is therefore zero.

We would like to view this system from frame K’, moving in the positive x-direction of K at constant speed v. In K’ the positive line charge is at rest and the negative line charge moves even faster toward negative x’. The test charge also moves to the left. B’ again points out of the xy-plane, and now the test charge feels a magnetic force upward. Yet, according to the force transformation the net force on the test charge must be zero in K’ as it is in K.

The solution to this conundrum lies in the fact that, according to the Lorentz transformation, the net line charge density is not zero in K’. The (now-resting) positive line charge density is less than it is in K, and the negative line charge density is greater than it is in K. The net line charge density in K’ is therefore negative, and there is an E’ field that points toward the current. Thus in addition to the magnetic force acting upward, the test charge experiences an electric force downward. This electric force quite elegantly cancels the upward acting magnetic force; the net Lorentz force in K’ is zero quite as it is in K.

Let us take a segment of the line charge in Fig. 3.1_1 and bend it into a square loop as depicted in Fig. 3.1_2. We produce an uncharged current loop. E=0 everywhere and there is now a dipolar B field.

Viewed from frame K’, the positive charge density in the bottom leg is less than it is in K, and the negative charge density is greater. The bottom leg thus has a net negative charge density in K’. The charge densities in the side legs are again zero. But the charge density in the top leg is positive. In K’ the translating current loop is electrically polarized. This dipole engenders a dipolar E’ field in K’. And of course B’ will not only be nonzero, but at any given point in space it will vary in time.

Fig. 3.2_1 depicts an accelerating, uncharged current loop. At any moment the degree of electric polarization is a function of the loop’s speed. As v increases in time, so does the electric polarization.

In Fig. 3.2_1 the magnitude of E at the loop’s center is increasing in time. This induces magnetic field components in the leading and trailing legs as indicated. And the current in each leg experiences a magnetic force opposite to the loop’s acceleration. The (unshown) agent causing the acceleration must counteract this magnetic force. The exerted force is proportional to a, and we may accordingly assign an electromagnetic mass to the loop such that (for v<<c)

. (3.2_1)

If the loop is one of the myriad microscopic loops in a disc-shaped magnet, then this inertial resistance to being (tangentially) accelerated will collectively contribute to the magnet’s overall moment of inertia. Evidently, given two discs, identical in every way except that one is magnetized and the other is not, the magnetized disc will have a greater moment of inertia. Among other things, a greater torque will be needed in order to attain a given angular acceleration.

Newton’s 2^nd law of course requires that a current loop also react with an inertial reaction force when it is radially accelerated. Fig. 3.2_2 depicts a current loop being forced to travel around a circle (at constant speed). Note that the loop itself rotates CCW as it travels along the circular path.

In this case, although the dipolar E field has constant magnitude, its direction varies as the loop travels along the curved path (rotating in the process). This nonzero dE/dt induces B field components as indicated in the left and right legs. In each case the resulting magnetic force is radially outward; it constitutes the loop’s inertial reaction force to radial acceleration. This force must be counteracted by the driving agent if the loop is to continue on the curved path.