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Re: Potential Energy of a Magnetic Dipole (small current loop)



"RAUBER, JOEL" wrote:

a) You introduce magnetic forces on moving charges and then on currents. If
you are like me you pound the table and make loud noises regarding the fact
that the magnetic forces *do zero mechanical work*.

b) Then we segue a few sections down the road in the textbook where we see
the discussion of the magnetic force and torque on a small current loop in a
*constant* uniform magnetic field. Naturally we calculate that the net
force is zero and that there is a net torque. Thus arriving at the "mu
cross B" formula for torque and then introducing a potential energy for the
work done by the "magnetic" torque we get the famous "- mu dot B" for the
potential energy of the current loop in the magnetic field.

If the magnetic forces do no work, how do we arrive at a potential energy
for the current loop in the magnetic field?


Robert Carlson replied:

Charges are moving within the loop.
The magnetic field applies a force to the moving charges,
attempting to make them follow a certain path. The
charges, in their attempt to follow this path, are
constrained to follow a different path by the constraint
of the wire and their electric attraction to it. In this
process, the constraint (the wire) applies a force on the
charges and the charges apply a force on the wire. It is
this force (the charges on the wire) that accelerates the wire.

That's exactly right.

The details are remarkably hard to visualize, so I made a
little drawing
http://www.monmouth.com/~jsd/physics/mag-loop.gif
which shows how the TOTAL force has a nonzero projection
on the TOTAL velocity.

Imagine that Joel's current loop consists of a positive
charge in the center, not moving, plus an electron tethered
to it by a rigid rod, orbiting around a hinge in the center.
The orbital motion of the electron _is_ the current loop.
Now suppose we want to change the plane of motion by flipping
it around the indicated axis. The magnetic field is
perpendicular to the page. Consider the instant, as shown,
when the loop is almost perpendicular to the page; this is
convenient because it means the force- and velocity-vectors
lie more-or-less in the plane of the page.

The magnetic force is perpendicular to the total velocity,
but the total force is not.

Work is done. Work is done against whatever mechanism is
attached to the hinge to maintain constant current.

Another point worth a certain amount of table-pounding:
Students often get the idea that a force of constraint
does zero work. That's not quite right. A _stationary_
constraint does zero work. Moving constraints can do
plenty of work.