Deep Dive: Work Done by Magnetic Forces

The Work-Energy Theorem and Magnetic Forces

It is a peculiar and fundamental property of the magnetic force that it does no work on a moving charged particle. At first glance, this assertion often provokes skepticism—after all, magnetic cranes lift cars, and electric motors drive machinery. However, if we examine the fundamental definition of work alongside the Lorentz force law, the conclusion is inescapable.

Let us consider a particle of charge Q moving with velocity v through a magnetic field B. The magnetic force exerted on this particle is given by the cross product:

\mathbf{F}_{\text{mag}} = Q(\mathbf{v} \times \mathbf{B})

Mathematical Derivation

Recall that the infinitesimal work dW done by a force F acting on a particle as it moves by a displacement dl is defined as the dot product of the force and the displacement:

dW = \mathbf{F}_{\text{mag}} \cdot d\mathbf{l}

Because the particle is moving, we can express the infinitesimal displacement in terms of its velocity and the infinitesimal time interval dt, such that \( d\mathbf{l} = \mathbf{v} \, dt \). Substituting the Lorentz force expression into our work equation, we obtain:

dW = [Q(\mathbf{v} \times \mathbf{B})] \cdot (\mathbf{v} \, dt)

We can rearrange the scalar terms to highlight the vector operations. The crucial step relies on the geometric definition of the cross product. The vector \( (\mathbf{v} \times \mathbf{B}) \) is, by definition, perpendicular to both v and B. Consequently, the dot product of this vector with v must be zero.

dW = Q [(\mathbf{v} \times \mathbf{B}) \cdot \mathbf{v}] \, dt = 0

Since the derivative of the work is zero, the total work done by the magnetic force over any path is identically zero:

W = \int \mathbf{F}_{\text{mag}} \cdot d\mathbf{l} = 0

Physical Interpretation

This result dictates that a static magnetic field cannot alter the kinetic energy of a charged particle. Since the work-energy theorem states that \( W = \Delta T \), and \( W = 0 \), the speed of a particle in a vacuum remains constant under magnetic influence alone.

The "Magnetic Crane" Paradox

If magnetic forces do no work, how can a magnetic crane lift a car? At first glance, it seems the magnetic force is doing work against gravity. However, a deeper look into the microscopic level reveals that the magnetic force is merely an intermediary.

The magnetic force acts on the charge carriers (electrons) within the conductor, pushing them sideways. These electrons then collide with the metal lattice of the crane's material, exerting a normal force that physically moves the crane and the car. While the magnetic force redirected the energy, the actual work of "lifting" is performed by whatever maintains the system's current or structural integrity, or by the electric fields produced by induction if the field is changing.

Internal Guidance

The magnetic force acts strictly as a guidance force—it is always pushing perpendicular to the motion, never along it. In an electric motor, the magnetic force acts on the wire segments, but the energy provided to rotate the motor ultimately comes from the battery (electrical potential energy), not the magnetic field itself. The battery must work harder to push the current through the wire against the "back EMF" induced by the motion.