Question

A point charge \( q \) is moving with velocity \( \mathbf{v} \) in a uniform magnetic field \( \mathbf{B} \). Find the work done by the magnetic force on the charge.

Hint:

The magnetic force on a charged particle does no work since it acts perpendicular to the velocity of the particle, causing no change in its kinetic energy.

Answer 1

The work performed by a force is defined as: \[ W = \mathbf{F} \cdot \mathbf{d} \] Here, \( \mathbf{F} \) represents the force and \( \mathbf{d} \) denotes the displacement. The magnetic force experienced by a moving charge is expressed as: \[ \mathbf{F} = q \mathbf{v} \times \mathbf{B} \] Given that the magnetic force is consistently perpendicular to the particle's velocity, the work done by this force is zero. This can be shown as: \[ W = \mathbf{F} \cdot \mathbf{d} = 0 \] Consequently, a magnetic force does not perform any work on a moving charge, thereby leaving the kinetic energy of the particle unchanged.