250 Week 7: Sources of the Magnetic Field
(and ideally, learns enough about them to incorporate them into ourgeneral picture of physics)
Gauss’s Law for magnetism will tell us that magnetic field lines produceno net flux through a closed
surfaceSand consequently must form closed loops in space.
The Biot-Savart Law for currents tells us how to compute the magnetostatic field produced
by a steady-state current distribution, if we can manage the complexity of dealing with vectors,
cross-products, and multivariate integral calculus simultaneously.
The “Heaviside” form for the magnetic field of a point charged particleqtravelling at some
velocity~v, although it is consistent with the Biot-Savart Law led us to some serious puzzles, enough
to make us doubt the consistency of classical physics itself. For one thing, we were able to show that
the interaction forces between two charged particles interactingwith this field violated Newton’s
Third Law and hence the Law of Conservation of Momentum for the pair! For another, Biot and
Savart only obtained their experimental Law by studying steady state currents, and a charged
particle exists only at a single point in space and isn’t smeared out into a“continuous” current; we
assumed that the magnetic field propagatesinstantaneouslyfrom the moving charge in the form we
wrote down, and as it will turn out, this is incorrect.
Finally, we obtained from the Biot-Savart Law a new equation we calledAmpere’s Lawafter its
discoverer that isconsistentwith it (one can derive the Biot-Savart Law from Ampere’s Law and,
with some effort, vice versa) but that inherits itsflawthat it is essentially a static result. We did
find Ampere’s Law to be remarkably useful for finding thestaticmagnetic field produced by suitably
symmetricstaticcurrent distributions, but we are, or should be, a bit worried aboutconsistency
because (hint hint) the “current through the closed curveC” that it explicitly references seems as
though it can mean nothing but the flux of the current density through some open surfaceSbounded
by that closed curve, but there are an infinite number of these surfaces and we (should) have the
uncomfortable feeling that the current we obtain stilldepends on the surface chosenwhere it really
shouldn’t.
Aninvariant form of the current – one that one could prove doesnotdepend on the surface
chosen – would be much better, especially if it still gives us the usual static result where it should,
but what physical principles might lead us to such an invariant form?
Ah, puzzles in abundance! Things are finally getting interesting! Thisis agoodthing, as reality
is undeniably rather complex and if the electric and magnetic force weretoosimple they could not
sustain the complexity we see every time we, well,see. This seems like a good time towrap up
electrostatics and magnetostaticsand move on to electric and magnetic fielddynamics.
We’ll begin by trying to understand a puzzle that we haven’t really faced until now. Magnetic
forces are by definition always exerted at right angles to the direction of motion of a charged particle
or moving current. This means thatmagnetic forces do no work!, because work requires a force
componentinthe direction of motion. Next week we will study what at first glance then seems like
a paradox – cases where magnetic fields clearly appear to do work – and thenresolvethe paradox by
concluding instead that magnetic fields under some circumstances createelectricfields, and electric
fields have no difficult at all doing work on charged particles!