Biot-Savart and Ampere’s Law
G.R.Dixon, 7/17/2005
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,
. (1)
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).
Figure 1
Uncharged Current Loop
Biot-Savart is used to compute B at points on the contour:
. (2)
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
, (3a)
, (3b)
. (3c)
B(0,y,z) is found by integration of Eq. 2:
. (4)
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
. (5)
Figure 2
Circular Contour
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
Figure 3
B dot dl Around the Contour
Despite the asymmetry of B dot dl in Fig. 3, it is found that Eq. 5 is satisfied:
, (6a)
. (6b)
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.