How do magnetic fields do work indirectly through electric fields?

Magnetic fields do work indirectly by causing changes in electric fields, which then exert forces on charged particles.

Magnetic fields themselves do not directly do work on charged particles. This is because the force exerted by a magnetic field on a charged particle is always perpendicular to the direction of motion of the particle. As work is defined as the force applied along the direction of displacement, the work done by a magnetic field on a charged particle is zero.

However, magnetic fields can do work indirectly through electric fields. This happens when a magnetic field changes over time. According to Faraday's law of electromagnetic induction, a changing magnetic field can induce an electric field. This induced electric field can then exert a force on charged particles, causing them to move. Since the force exerted by the electric field can be in the same direction as the motion of the charged particles, the electric field can do work on the charged particles.

For example, consider a loop of wire in a changing magnetic field. The changing magnetic field induces an electric field in the wire. This electric field exerts a force on the free electrons in the wire, causing them to move. This movement of electrons constitutes an electric current. The electric field does work on the electrons, transferring energy to them. This energy is then dissipated as heat or used to do work in an electrical device connected to the wire.

In this way, a magnetic field can indirectly do work on charged particles by inducing an electric field. This principle is fundamental to the operation of many electrical devices, including electric generators and transformers. In these devices, a changing magnetic field induces an electric field, which then does work on charged particles, causing an electric current to flow.