electromagnetism - How can a magnetic field accelerate particles if it cannot do work?

A varying magnetic field can accelerate charge particles, but it is said that a magnetic field can't do any work so it should not be able to speed up charged particles, right? How is this apparent contradiction resolved?

Answer

A varying magnetic field generates an electric field, and an electric field can do work on a particle. This is called Faraday's law of induction:

$$\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$$

The full Lorentz force equation is

$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$

So for example, if the magnetic field is increasing in the $\hat{z}$ direction, such that

$$\vec{B} = b t \hat{z}$$

and

$$\frac{\partial \vec{B}}{\partial t} = b \hat{z}$$

then the electric field is determined by

$$\nabla \times \vec{E} = - b \hat{z}$$

Thus the electric field is not zero, so work can be done on a charged particle as a result of a changing magnetic field.