Didactical formulation of the Ampère law

Abstract

The Ampère law is useful to calculate the magnetostatic field in the cases of distributions of current with high degree of symmetry. Nevertheless the magnetic field produced by a thin straight wire carrying a current I requires the Biot–Savart law and the use of the Ampère law leads to a mistake. A didactical formulation of the Ampère law is proposed to prevent misinterpretations.

The magnetic field circulation counterpart to Biot-Savart’s law

“…By contrast to approach 1, approach 2 has been extensively used for a finite length straight wire, e.g. [2,3,7,9,[14][15][16][17]. The two approaches are next applied to a volume distribution which approximates a thin arbitrarily shaped wire.…”

“…configurations for which the current has to be closed in order to satisfy the requirement of continuity (charge conservation), e.g. [2,3]. By contrast, Biot-Savart's law holds in circumstances more general than this, which are detailed in [4].…”

“…direct applicability in high-symmetry problems which involve determination of the magnetic field resulting from steady flow of current through a finite portion of conductor. Moreover, the finite conductor problem has been identified as one of sources of student difficulties regarding the correct application of Ampère's law [2,3,7], which is a relevant topic in physics teaching [8]. Recently, [9] used the integral form of Ampère-Maxwell's law in conjunction with Coulomb's law in the zero retarded-time limit to obtain a simple formula that allows the circulation, around an arbitrarily shaped loop, of the magnetic field generated by a current in an arbitrarily shaped finite wire, to be expressed in terms of the wire current and of the solid angles subtended by the circulation path and each of the wire's endpoints.…”

Comment on ‘Didactical formulation of the Ampère law’

“…holds for all (proper) surfaces and boundaries due to Stokes's integral theorem. The suggested constraint on the boundaries in [1], is always fulfilled for any surface due to topological reasons, if the charge-conservation constraint (3) is fulfilled: Indeed, for a finite current distribution in some arbitrarily shaped wires it implies that the wires must form closed loops in order to fulfill it. Then the constraint on the surfaces given in the paper is always fulfilled: Any surface bounded by a closed loop that encircles one wire must contain one point of the wire, implying that the integration of the current density over any surface leads to the same result, namely the total current running through that wire.…”

“…In [1] a straight wire along the z axis of length L (−L/2 < z < L/2) is considered. Assuming a constant total current I is running in this finite piece of wire, the current density is given by…”