Nonequivalence of the magnetostatic potential energy corresponding to the Ampère and Grassmann current element force formulas

Abstract

The equivalence of the Ampère and Grassmann (Biot–Savart/Lorentz) current element force formulas is well established. However, when the magnetostatic potential energy corresponding to these force formulas is evaluated, the formulas are found to be nonequivalent. The historical current element force formulas are first presented. The magnetostatic potential energy corresponding to these historical current element force formulas are then analysed. The end result establishes the Grassmann and Neumann current element force formulas as the only commonly referenced formulas giving the correct magnetostatic potential energy for circuital currents.

A pictorial view of the relationship between magnetic flux and the inductance terms of two coaxial current loops in free space

“…. As such, the differential element of magnetostatic potential energy corresponding to the general differential current element force formula of (36) could just as well be: and the magnetostatic potential energy corresponding to the Moon and Spencer differential current element force formulas of (37) could just as well be: Using a similar process found in prior work of computing the magnetostatic potential energy between various closed loops [43], the constraints on k 1 , k 2 , and k 3 of ( 36), ( 37), (42), and ( 43 , the end result is an attractive force between the two current loops of figure 1 and the correct positive valued mutual magnetostatic potential energy associated with the two loops.…”

“…For magnetostatics, the differential element of force on a differential current element of the second current loop (the top loop of figure 1) interacting with a differential current element of the first current loop can take on an infinite number of possibilities [43] (see derivations by Whittaker [44], OʼRahilly [45], Stefan [46], and Moon and Spencer [47]): where k 1 and k 2 are arbitrary constants. By maintaining Ampèreʼs constraint that current element forces only act along the straight line between them, but removing Ampèreʼs original constraint that a current element canʼt have a tangential force component (when interacting with a closed circuit) [48], Moon and Spencer derived an additional infinite formula set for the force between differential current elements [47]: where k 1 and k 3 are arbitrary constants.…”

“…2 by computing the magnetostatic potential energy between various closed loops [43], yielding the Grassmann (Biot-Savart/Lorentz) current element force formula [49,50]: Simliarly, prior work established the constraints on k 1 and k 3 of ( 37) and (39) as = k 1 1 and = k 0…”

“…3 by computing the magnetostatic potential energy between various closed loops [43], yielding the Neumann current element force formula [51]:…”

A magnetic vector potential corresponding to a centrally conservative current element force

“…The actual force on a differential current element can not be determined experimentally because isolated differential current elements do not physically exist [10]. The differential element of force on a static differential current element of one current loop (contained in the differential volume dV) from the interaction with a differential current element from another current loop (contained in the differential volume ′ V d ) can take on an infinite number of possibilities [11] (see derivations by Whittaker [12], OʼRahilly [13], Stefan [14], and Moon and Spencer [15]): where ⇀ r is the location of the differential current element…”