232 Week 7: Sources of the Magnetic Field
rB (out of page) B (into page)qvθFigure 77: The geometry of the magnetic field lines going in circlesaroundthe (dotted) line of
motion of the charge in the right-handed sense. Note that the direction of~v×~ris into the paper
on the right, out of the paper on the left, for~vand any of the~rvectors shown.
For the field “near” a more or less stationary (slowly moving) electriccharge this didn’t matter
much, but for a moving charge (where there is no magnetic field at allwithout the motion) itstarts
to matter, and indeed advanced students may wish to ask themselves whathappensto the magnetic
field if we change our point of view to an inertial reference frame that is moving with the velocity
of the charge. Hmmm, it seems as though inthisframe thereis no magnetostatic fieldwhere in the
frame where the charge was moving, there was one! Yet somehow,thephysicsobserved must be
the same! This will take a bit of work, in some future E&M course, to arrange, and the theory that
consistently permits it all to work out is thetheory of special relativity.
Violation of Newton’s Third Law
Another problem that we can understand rightnowfollows from combining this empirical law for
building the field with the Lorentz force law for the magnetic force ona moving charge. Put them
together and we find that for two chargesq 1 andq 2 travelling at velocities~v 1 and~v 2 and with~r 12
the vector fromq 1 toq 2 , the force onq 1 due toq 2 is:
F~ 12 =−q 1 ~v 1 ×kmq 2(
~v 2 ×rˆ 12
r^212)
=−
kmq 1 q 2
r^212(~v 1 ×(~v 2 ׈r 12 )) (499)Similarly, the force onq 2 due toq 1 is:
F~ 21 =q 2 ~v 2 ×kmq 1(
~v 1 ×rˆ 12
r^212)
=
kmq 1 q 2
r^212 (~v^2 ×(~v^1 ×rˆ^12 )) (500)There is just one wee problem with this result. F~ 126 =F~ 21! Newton’s Third Law has just
bitten the dust, never to return! It isnot correctthe way it is usually taught, and its failure has
profound implications! In case the inequality of these two terms (except for a minus sign) isn’t
obvious, consider the geometry in figure 78: As you can easily see byinspection, the magnetic field
at the position ofq 2 in this figure is zero, because~v 1 isparallelto~r 12 (so the cross product is zero).
However,~v 2 is perpendicular to~r 12 and the cross-product is not zero; the magnetic field atq 1 is
nonzero and in to the page, perpendicular to~v 1. Consequently the force onq 1 is nonzero and points
to the left while the force onq 2 is zero!
If you think for a moment, you will recall that weusedNewton’s Third Law to derive a very
important physical principle: The Law of Conservation of Momentum! In the picture above, the
total momentum of the interacting pair of particles ischanging in time. If we worked harder and
computed the way the totalenergyof the particle pair is changing as a function of time we would
find that it is changing too. The pair of particles is literally “lifting itself up by its own bootstraps”!