260 Week 8: Faraday’s Law and Induction
The charge in the rod thereforepolarizes, creating a net negative charge at one end and a net
positive charge at the other end that create anelectric field in betweenpointing from the top end
to the bottom one. Charge will move until the remaining free chargein the rod in between the
ends experiences no net force when the electric and magnetic forces balance. The rod spontaneously
forms a region of crossed fields, exactly the same way it spontaneously formed in the case of the Hall
Effect, only now there is no current; the forces that balance are brought about solely by the motion
of the rod through the stationary, uniform magnetic field!
We can easily deduce the condition for force balance for the charges in the rod proper:F~m+F~e= 0 (559)or (since they are in opposite directions and the motion is at right angles to the magnetic field)
qvB=qE (560)or the magnitude of the electric field that is generated in the polarized rod is given byE=vB. This
field, in turn, creates anelectric potential differencebetween the ends of the rod:
∆V=L·E= (vL)·B (561)If we were to somehow construct a conducting pathway between the ends of the rod, we would
expect current to flow, and naively at least we would expect it to be driven by the magnetic force
on the charges even though we know that they cannot be doing anywork. This leads us to a bit of
a paradox – if the magnetic field isn’t doing any work, what is?
To answer this, we note that we are examining what happens in a frame of reference in which the
rod moves through a static magnetic field. Let’s imagine that we havejumpedonto the rod so that
we are now at rest and the magnetic field is sweeping past us theopposite way. In this case we have
no reason to think that there should be a magnetic force on charges in the rod at all! They are all
at rest in the frame of reference we are in, and the magnetic field they are moving in isn’t varying,
it is constant in magnitude and direction! Yet things like the observeddistribution of charge in this
stationary frame has toagreewith the distribution in the frame in which the rod moves, because
physical reality itself cannot change along with our point of view; thecharges are where they are
(at the ends of the rods) no matter the frame we look at the rod in.Even in this stationary frame,
then, the charge in the rod has apparently polarized and generates thesameinternal electric field
between the charges at the ends that we saw in the moving frame.
If there is no possible way for a magnetic force to be exerted on thestationary charges in the rest
frame of the rod, the only remaining force that the charges can see is an electric force. A consistent
explanation, however odd it might seem at first, is that the motion ofthe rod through the magnetic
field, when viewed in the frame of the stationary rod, has generated anexternal electric field from
the bottom of the rod towards the top!This field has actedexactlylike an external field always does,
and created surface charge densities at the ends that polarize the rod until the internal fieldcancels
the external field inside of the conductor!
Because our results for the reaction/polarization field have to agree in both frames (where elec-
trostatic fields shouldn’t depend on the frame) the “induced” external fieldE~indmust be equal in
magnitude and opposite in direction to the polarization field:
Eind=vB (562)but pointingup, not down, when seen in the rest frame of the rod. This, believe it ornot, is our
first glimpse of a natural law that is one of the fundamental cornerstones of human civilization in
disguise – without it our lives would be far, far poorer.
By once again using our imagination to change our point of view to a different inertial reference
frame and using the expected invariance of the laws of physics whenwe perform such a change in