Given that the loop of wire is falling down and the magnetic flux is changing, a current is induced in the counter-clockwise direction.
When calculating the force on the wire, $ Ids \times B $, it points in the $-\hat{j} $ for the upper part of the wire. And $\hat{i} $ and $-\hat{i} $ for the left and right side respectively and they cancel each other. The bottom side is out of the magnetic field so there is no force in there.
When it comes to determining the work done by the magnetic force in the falling loop my rationale was that both forces on the sides are perpendicular to the displacement of the loop in the $\hat{j} $ direction so they do no work, and the upper force is opposite to the displacement of the loop so the work done is negative.
However, the correct answer, apparently, is that magnetic forces do not do work as they are always perpendicular to the displacement. I understand that magnetic forces do not work on individual charged particles.
However, this contrasts with my "macroscopic" argument of the wire being "pulled up" while the displacement is down, i.e. negative work.
Where's my understanding wrong? Does the magnetic force do work or not?
Best Answer
Your understanding is correct, macroscopic magnetic force (acting on a current-carrying conductor in external magnetic field) in general can do work on the conductor. It does positive work when the conductor moves in direction of the magnetic force, and negative work when the conductor moves in direction opposite to the magnetic force. It's just like work of any other macroscopic force.
The statement "magnetic force does not work" is correct primarily in the special case where the "magnetic force" is actually the magnetic component of the Lorentz force acting on a moving charged point particle in vacuum, which is often expressed as
$$ \mathbf F_{magnetic~part~of~the~Lorentz~force} = q\mathbf v \times \mathbf B , $$
where $q$ is electric charge of the particle, $\mathbf v$ its velocity and $\mathbf B$ is external magnetic field.
This magnetic component of the Lorentz force on a single charged point particle in vacuum is always perpendicular to velocity of the particle, so the work done by it is always zero.
However, in macroscopic electromagnetic theory, "magnetic force on conductor" refers to ponderomotive force (force acting on heavy mass, not on mobile charge carriers), sometimes also called "motor-action" force, or Laplace or Ampere force in French sources. The simplest familiar case of this is magnetic force acting on straight current-carrying wire in external magnetic field. Its magnitude is
$$ \mathbf F_{macroscopic~magnetic} = I\mathbf L \times \mathbf B $$ here $I$ is electric current and $\mathbf L$ is vector whose magnitude is length of the wire and direction is that of the current. This formula depends on electric current, but it does not depend on velocity of the conductor; the latter is assumed to be zero or low enough so that it does not matter (if the conductor moves with very high velocity, this may affect current $I$, but this is usually neglected).
This force acts on the whole body of the conductor and because the conductor moves, in general, with different velocity than mobile charge carriers do, this macroscopic force is not in general perpendicular to velocity of the conductor. So work of this force is not in general zero and the conclusion about zero work from the simple example with single particle above does not apply. We are simply dealing with different kind of "magnetic force" here. There are other similar examples with permanent magnets, electromagnets etc. Magnetic force they exert on other bodies can do work on them. The most useful familiar case is the electric motor - inside, macroscopic magnetic forces do work on the rotor (because it has either moving conductors, or moving magnets).
In your example, magnetic force does negative work on the circuit as it moves down, and thus extracts energy from it. This energy goes to magnetic energy of induced current and some small amount gets radiated out, and then later the magnetic energy dissipates into heat and further small amount of radiation as the current decays to zero.