Question

What is the work done by an external uniform magnetic field perpendicular to the velocity of a moving charge?

Hint:

A magnetic field can change the direction of a charged particle's motion (making it move in a circle), but it cannot change its speed or kinetic energy because it does no work on it.

Answer 1

The work done by the magnetic field is zero. The magnetic force on a moving charge (Lorentz force) is given by \( \vec{F} = q(\vec{v} \times \vec{B}) \). By the definition of the cross product, the force \( \vec{F} \) is always perpendicular to both the velocity \( \vec{v} \) and the magnetic field \( \vec{B} \). Since work done is \( W = \vec{F} \cdot \vec{d} \), and the displacement \( \vec{d} \) is in the direction of velocity, the force is always perpendicular to the displacement. Therefore, the dot product is zero, and no work is done.