Biot-Savart and Ampere’s Law
Given an uncharged current loop whose center is at rest in an inertial frame, E=0 everywhere, and B is constant in time and specified by Biot-Savart. According to Ampere’s law,
Here the integral is around a closed contour, and I is the current threading the area spanned by the contour. The integral is positive if the thumb of the right hand points in the direction of I and the curled fingers point in the direction of positive dl.
The program featured in this article tests Ampere’s law for a circular current loop. Fig. 1 shows the loop, viewed from the +z-axis. The dots at (0,R-R/2,0) and (0,R+R/2,0) indicate a cross section through the contour of integration. More generally the contour is a circle of radius R/2 and lying in the yz-plane. It is centered on point (0,R,0).
Uncharged Current Loop
Biot-Savart is used to compute B at points on the contour:
Here r is the displacement vector from a point on the current loop to a point on the contour, and Rdq is tangent to the current loop and has the same direction as I. If (0,y,z) is the selected point on the contour, then
B(0,y,z) is found by integration of Eq. 2:
Having computed B(0,y,z) at points around the contour, we can integrate B dot dl and compare it to I/eoc2 (Eq. 1).
Fig. 2 shows the contour from the perspective of positive x. Note that the current travels in the negative x-direction at the contour’s center. Thus we should actually find that
It is noteworthy that B dot dl is not single-valued at points around the contour (as it is in the case of an infinitely long, straight-line current). Fig. 3 plots B dot dl vs. f for the contour in Fig. 2
B dot dl Around the Contour
Despite the asymmetry of B dot dl in Fig. 3, it is found that Eq. 5 is satisfied:
The software used in this article can readily be modified to compute B(x,y,z) at other points. For example, it is easily demonstrated that (barring numerical error) the integral of B dot dl around a contour not spanning I is zero. In brief, Ampere was right!
If a second current loop is substituted for the contour of choice, then the Lorentz magnetic force can be used to compute the net force and/or torque on this second loop. And if both loops are held at rest, then any net force/torque on the second loop is matched by an equal, oppositely directed force/torque on the first.
More generally it is the ambient field that exerts a force/torque on a given current loop, and the loop’s equal and oppositely directed force/torque acts back upon the field. If the loop(s) is (are) free to accelerate (linearly and/or angularly), then the field will vary in time in accordance with Maxwell’s equations. In the case of unconstrained loops, at any given moment the force/torque on one loop might not equal the negative force/torque on the other. But Newton’s third law is nonetheless satisfied. For in general it is a current loop and its ambient field that constitute the action/reaction entities. And the forces/torques between an interacting loop and its ambient electromagnetic field constitute a Newtonian action/reaction pair.