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 R__d____q__ 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/e_{o}c^{2} (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.