Frank's apparent objection to my observation that classical magnetic
fields do no work got me thinking ... and the consequence is a cute
little puzzle which some in the group might find amusing.
Let's imagine a small experiment with a couple of _ideal_ objects. I am
telling you up front these objects obey Maxwell's equations, and
Newton's laws of mechanics (or Einstein's laws if you want to be picky);
there's no wiggle room for any uncertain "reality" to sneak in!
First object: A black box which produces a perfect, unvarying magnetic
dipole field. The box is anchored to the table and the table is
anchored to the earth. You don't need to know what's inside it; only
its field is interesting.
Second object: A rigid ring of uniform charge density. The ring has
mass M, radius R, total charge Q; the charge is evenly distributed
around the ring.
The charge in the ring obeys the Lorentz force law: the force on each
infinitesimal portion of the ring will be q(E+VxB) where E and B are the
electric and magnetic fields in space at that point on the ring, q is
the (infinitesimal) charge contained in that infinitesimal portion of
the ring, and V is the velocity of that particular point on the ring.
Set the ring to spinning, at angular velocity W. (Now the ring is
behaving just like a current loop, of course.) Place it near the box,
oriented so that the magnetic dipole field the spinning ring creates is
oriented opposite to the dipole generated by the box. (Like two bar
magnets, with N poles lined up with S poles.)
A moment's fiddling with the right-hand rule should be enough to
convince you that the ring is _attracted_ to the box (keep in mind that
the field from the black box falls off with distance).
Now, let the ring go. ZZZIIIP! it heads for the box ... BAM! it hits
the side of the box.
Oh, looky there -- the collision dented the box!
Hey, wait a minute -- denting the box took _energy_. But ... magnetic
fields _do_ _no_ _work_. That's obvious from the Lorentz force law, and
in this idealized case there is no wiggle room: we have specified that
the law Must Be Followed.
So, where'd the energy to dent the box come from?
;-)
