Magnetic Forces Doing Work?

Abstract

Consider a conducting circuit moving with velocity v through a constant magnetic field B. The induced emf is given by ∫ v × B · dl where the integral is taken once around the circuit. Some texts refer to v × B as the force the magnetic field exerts on a unit charge moving around the circuit. This is incorrect as magnetic forces can never do work. The force the conductor exerts on an electron is shown to do this work.

The Hall effect in accelerating and stationary conductors

“…In some textbooks 2,3 and articles 4,5 it is mentioned that the internal electric field needed to accelerate a conductor carrying current in a magnetic field is the Hall effect field. Rostoker 6 provided a quantitative derivation of the Hall field, based on the Bloch-Sommerfeld theory of conductivity, for conductors constrained to be at rest, with the tacit assumption that no forces of constraint act directly on the electrons.…”

Does the electromotive force (always) represent work?

“…The balance of forces and the origin of work in a conducting circuit moving through a magnetic field are nicely discussed in [2,8,9]. An interesting approach to the relation between work and emf, utilizing the concept of virtual work, is described in [10].…”

“…Note that this latter diagram concerns only the moving part (ab) of the circuit, since it is in this part only that the velocity υ and the applied force app f are nonzero. The emf of the circuit at time t is, according to (8),…”

“…A uniform magnetic field B , pointing into the page, fills the entire region. The emf of the circuit at time t is, according to (8),…”

The role of the virtual work in Faraday's law

“…(1), is carried out at a fixed time. Therefore, it does not correspond to a real work, since a real work must consider time-evolution [4,11]. In our subsequent developments, we shall show the following (a) Equation ( 1) can also be interpreted as a virtual work done by the magnetic force f on the charge q.…”