On the Role of Translating Current Loop Electric
Polarization in the Field Transformations
Fig. 1 depicts the middle section of an infinitely long, rectangular current loop. The legs shown are joined at + infinity by short, vertical legs of length d. The loop consists of superimposed positive and negative line charges, each with a charge density of magnitudel. The positive charge is at rest and the negative charge circulates clockwise at speed v. Thus there is a current of
Since the net charge density is zero, there is no electric field. And at all points on the x-axis B points in the positive z-direction.
Section of Rectangular Current Loop
At the Origin the magnetic contribution of either segment ismoI/2p(d/2), and thus the total magnetic field is
Let us view things from frame K’, moving in the positive x-direction at speed v. In K’ the negative charge in the top leg is at rest:
The negative charge in the bottom leg has a velocity of
Furthermore, the positive charge in each leg moves in the negative x’-direction:
The charge densities are also different. The negative charge in the top leg has its least density in frame K’ (its rest frame):
In the bottom leg,
The net charge densities in K’ are therefore
In frame K’ the loop is electrically polarized.
At the Origin of K’, the magnetic fields of the top and bottom positive charges cancel. And of course the top negative charge, being at rest in K’, has no magnetic field. Thus Bz’ is solely due to the bottom negative charge, which constitutes a current of
quite as the field transformation specifies.
Also, there is an electric field in K’:
again in agreement with the field transformation.
It isn’t difficult to compute the polarization of charge for other loop shapes when they are in motion. This was done for a circular (in frame K) loop in a previous article. Key to such exercises is a determination of where all of the loop’s charge increments are at a single instant in K’. Such a determination entails some post-transformation treatment, since the basic Lorentz transformation specifies where different charge increments are at different moments in K’. While the added treatment is simple when all objects are at rest in K, the process is slightly more involved when some of the objects move in K (as the negative charge increments do in the simple current loop model).
It is interesting to note what happens if multiple current loops, like the one depicted in Fig. 1, are stacked in the xy-plane. In this case the currents, in the superimposed legs of adjacent loops, cancel and only the currents in the top- and bottom-most legs persist. Note in Eqs. 10 and 11 that the net charge densities in all internal legs also sum to zero. If an infinite number of such loops are stacked, all the way out to + infinite y, then the fields vanish entirely at all points a finite distance from the Origin.
The same would be true for finite length, rectangular current loops, arranged in infinite length rows. Philosophically speaking, "empty space" could be populated with infinite arrays of such loops. There is no way we could detect their existence! That is, they would manifest themselves with nonzero fields only at places where the "fabric" was compromised, displacing the legs of neighboring loops so that they no longer overlapped.