electromagnetism - How can magnets be used to pick up pieces of metal when the force from a magnetic field does no work?

I learned that the force from a magnetic field does no work. However I was wondering how magnets can be used to pick up pieces of metal like small paperclips and stuff. I also was wondering how magnets can stick to pieces of metal like a refrigerator?

Answer

The Lorentz force $\textbf{F}=q\textbf{v}\times\textbf{B}$ never does work on the particle with charge $q$. This is not the same thing as saying that the magnetic field never does work. The issue is that not every system can be correctly described as a single isolated point charge

For example, a magnetic field does work on a dipole when the dipole's orientation changes. A nonuniform magnetic field can also do work on a dipole. For example, suppose that an electron, with magnetic dipole moment $\textbf{m}$ oriented along the $z$ axis, is released at rest in a nonuniform magnetic field having a nonvanishing $\partial B_z/\partial z$. Then the electron feels a force $F_z=\pm |\textbf{m}| \partial B_z/\partial z$. This force accelerates the electron from rest, giving it kinetic energy; it does work on the electron. For more detail on this scenario, see this question.

You can also have composite (non-fundamental) systems in which the parts interact through other types of forces. For example, when a current-carrying wire passes through a magnetic field, the field does work on the wire as a whole, but the field doesn't do work on the electrons.

When we say "the field does work on the wire," that statement is open to some interpretation because the wire is composite rather than fundamental. Work is defined as a mechanical transfer of energy, where "mechanical" is meant to distinguish an energy transfer through a macroscopically measurable force from an energy transfer at the microscopic scale, as in heat conduction, which is not considered a form of work. In the example of the wire, any macroscopic measurement will confirm that the field makes a force on the wire, and the force has a component parallel to the motion of the wire. Since work is defined operationally in purely macroscopic terms, the field is definitely doing work on the wire. However, at the microscopic scale, what is happening is that the field is exerting a force on the electrons, which the electrons then transmit through electrical forces to the bulk matter of the wire. So as viewed at the macroscopic level (which is the level at which mechanical work is defined), the work is done by the magnetic field, but at the microscopic level it's done by an electrical interaction.

It's a similar but more complicated situation when you use a magnet to pick up a paperclip; the magnet does work on the paperclip in the sense that the macroscopically observable force has a component in the direction of the motion of the paperclip.