The Fields of a Translating, Uncharged Current Loop
G.R.Dixon, 7/14/2005 (Revised 2/2/2006)
Given a complete knowledge of E(x,y,z,t) and B(x,y,z,t) in inertial frame K, E’(x’,y’,z’,t’) and B’(x’,y’,z’,t’) in inertial frame K’ can always be determined in the following way: (1) For any (x’,y’,z’,t’) use the Lorentz transformation to determine (x,y,z,t); (2) Subject E(x,y,z,t) and B(x,y,z,t) to the general field transformations in order to determine E’(x’,y’,z’,t’) and B’(x’,y’,z’,t’).
This procedure is simplified if E(x,y,z) and/or B(x,y,z) are independent of time in K. For example, when (a) the center of a current loop is permanently at rest in K and (b) the circulating charge’s motion is non-relativistic, then Coulomb specifies the electrostatic field and Biot-Savart specifies the magnetostatic field. Regardless of what value of t a given t’ transforms to in such cases, the general field transformations can be applied to E and B to obtain E’ and B’. For convenience the general field transformations are provided below. The assumption is that K’ moves in the positive x-direction of K at speed v.
In this article the magnetic and electric fields of an uncharged, translating current loop are computed at selected points for two cases: (1) the current loop moves with a constant velocity normal to its plane, and (2) the current loop moves with a constant velocity parallel to its plane. (The software that computes the fields can easily be modified to compute the fields at other points of the user’s choosing.) Inertial frame K is the frame in which the loop’s center is permanently at rest at the origin. In both K and K’ the loop’s total charge is zero. In K the loop is circular, with a radius of R = 1 meter, and the charge density is everywhere zero. This is also true when the loop moves perpendicular to its plane. The constant current is I = 1 amp and it is assumed that the speed of the circulating charge in frame K is non-relativistic.
In Case 1 the loop lies in the yz-plane, and in Case 2 it lies in the xy-plane. Relative to inertial frame K’, the loop’s center travels in the negative x’-direction with the constant, mildly relativistic speed v = .75c. In both cases the fields are computed at the single instant t’=0, which is when the loop’s center coincides with the origins of both K and K’. In Case 1 the field evaluation points are on or above the x/x’ axis and the range of x’ is –3R<x’<3R. In Case 2 the field evaluation points are on the z/z’-axis and the range of z’ is –3R<z’<3R. Since the loop is uncharged, E = 0 everywhere in K and B can be computed using Biot-Savart:
Here dl is an infinitesimal segment of the loop, whose direction is the same as the flow of I, and r is the displacement vector from that segment to field evaluation point (x,y,z).
2. Case 1: Loop Translates Perpendicular to its Plane.
In this case the loop lies in the yz-plane of frame K. Viewed from positive x the current circulates counterclockwise. E is everywhere zero, and at points on the x-axis B has only a positive x-component. Fig. 2_1 plots Bx’’(x’). (It has the same shape as a plot of Bx(x), but at any given instant in K’, K is length-contracted.)
In K’ the magnetic field varies in time and thus curl E’ is nonzero. Consequently the current loop has a nonzero E’ field at points off the x’-axis. At such points in the x’y’-plane Ex’’ and Ey’’ are zero, but Ez’’ is nonzero. For example, Fig. 2_2 plots Ez’’(x’,2R,0).
Since the charge density is zero everywhere in K’ (as it is in K), div E’ = 0 at all points in K’. The nonzero E’ plotted in Fig. 2_2 is therefore completely circulatory (or non-conservative); i.e. it is strictly induced by the rate at which B’ varies in time. The lines of E’ describe circles (a) centered on the x’-axis and (b) with planes that are parallel to the y’z’-plane.
3. Loop Translates Parallel to its Plane.
In this case the loop lies in the xy-plane of frame K. The current circulates counterclockwise, viewed from positive z. The loop translates parallel to its plane in the negative x’-direction of K’ at the constant speed v = .75c. As discussed in other articles, under these circumstances the current loop is electrically polarized in K’. (Furthermore, at any given moment in K’ the loop is length-contracted in the +x’-direction.) In particular the loop will have an excess of positive charge on its positive y’-side and an excess of negative charge on its negative y’-side. But note that although the charge density is not zero around the loop in K’, the total charge is still zero.
Whereas E’ was zero on the loop axis in Case 1, in this case there is a nonzero Ey’’ component normal to that axis. The sign of Ey’’ is consistent with the moving current loop’s translation-induced electric dipole moment. Fig. 3_1 plots Ey’’(0,0,z’) at time t’=0. (Ex’’(0,0,z’) and Ez’’(0,0,z’) are both zero in this case.)
4. Gauss and the Loop’s Electric Polarization in Case 2.
In the case of a line current (zero cross section and infinite current density) the magnitude of B approaches infinity as the distance from the current approaches zero. Furthermore, the direction of B abruptly reverses at the current. Eq. 1_1b thus indicates that at point (0,R+dy’,0) and at time t’=0, Ey’’ is positive. At point (0,R-dy’,0), Ey’’ is negative. The flux of E’ through a pillbox of thickness 2(dy’) and enclosing the increment of current is therefore positive. According to Gauss, there is a net positive charge within the pillbox. At points +(g-1R,0,0) E’ lies in the pillbox surfaces and the electric field flux is zero. The net density of the overlaid positive and negative line charges is evidently zero at these points. And of course at the loop’s "bottom" (y’<0) the flux through the pillbox is negative. The moving loop has a translation-induced electric dipole moment that points in the positive y’-direction … a result implied both (a) by Gauss and the general field transformations, and (b) by application of the Lorentz transformation to the positive and negative charge increments comprising the loop.
When an uncharged current loop moves perpendicular to its plane, it remains uncharged at all points in the loop. However, it has a non-conservative electric field that circulates around the line of motion.
The electric field of an uncharged current loop that moves parallel to its plane is consistent with the Lorentz transformation result that such a loop has a nonzero electric dipole moment when it translates. That is, although the loop’s individual charge increments may be invariant under a Lorentz transformation, the circulating increments are not evenly distributed around the loop at any given instant. The electric field of such a loop accordingly has both non-conservative and conservative components.
To the extent a permanent magnet can be modeled as an array of microscopic, uncharged current loops, similar electric polarization can be expected when the magnet translates perpendicular to its North/South axis. In such cases the magnet’s electric field will have both circulatory and divergent components. In effect the magnet has a nonzero electric dipole moment when it translates perpendicular to its North/South axis.
A special case occurs when a disc-shaped, uncharged permanent magnet spins around an axis through its center. Although none of the magnet’s theoretical microscopic current loops are at rest in any inertial frame under these circumstances, experiments indicate that here too there is electric polarization and a net conservative electric field. (The spin-induced electric field in this case has radial components; the motion-induced electric dipole moments in the individual, microscopic current loops point toward or away from the magnet’s spin-axis.)
It is noteworthy that this radial field does not exist when an uncharged current loop, such as those discussed in Cases 1 and 2 above, is spun around an axis through its center but does not as a whole translate. The motion-induced electric polarization of an uncharged current loop occurs only when the loop’s center translates. In the disc-shaped permanent magnet cited above, each of the theoretical constituent microscopic current loops translates when the overall magnet is spun, even though the center of the overall magnet remains at rest.