Week 8: Faraday’s Law and Induction 261
frame, we have discoveredinduction– the creation of electric fields by changing magnetic fields.
We have a ways to go before we completely understand this and can write the result down as our
fourth and final Maxwell equation,Faraday’s Law, but we can already see that it must be so as the
result beautifully resolves the paradox of “what does the work” onmoving charges in a magnetic
field (which can do no work, yet work as we shall see in a moment is clearly done).
In the next section we will reconsider this rod when we do indeed provide it with an idealized
conducting pathway that allows current to flow. In the process, we will get a step close to a suitable
general formulation of the underlying physical principle.
8.2: The Rod on Rails
x
R
I
B (in)
F
Lq v
Figure 92: A conducting rod of massMand lengthLmoving through a uniform magnetic field into
the page and sliding onfrictionless conducting railsthat are connected by a resistorRoutside of
the magnetic field. Current can flow around the loop thus formed.
In figure 92 we have added a pair of frictionless conducting rails connected by a wire outside of
the magnetic field. The total resistance of the loop thus formed (including the rod) isR. We have
added anxcoordinate to show indicate the instantaneous position of the rod,which is still moving
to the right at speedv.
In the previous section we decided that while in the lab it looked as though there was a magnetic
force acting up on any given free chargeqin the rod (which is now free to move all the way around
the loop as part of a “continuous” currentIformed in the usual coarse-grained limits we have now
seen several times), in the frame of the rod itself there was anexternal electric fieldgenerated as it
moved through the magnetic field of magnitudeE=vBthat is what actually pushes the charges
along, doing work as needed. Of course this electric fieldnowhas to exist in the entire conducting
pathway as it has to push the charges along against the actual resistanceR, and we know that to
properly ensure that the work-energy theorem is satisfied, we should think not of the field, but of
the potential difference produced by the field. The potential difference induced across the rod as it
moves is just:
∆Vind=EindL= (vB)L (563)
The last thing that has to trouble us is the sign of this potential difference. Again we need to
appeal to physical invariance – in both frames we know that the magnetic force or induced electric
force respectively must push the charges around the loopcounterclockwisewhenvis to the right.
Since we want Kirchoff’s Loop Rule to be satisfied for this simple circuit loop and the voltage
decreasesacross when we move across the resistor byIR, we expect the voltage around the loop to
be positive, so that:
∆Vind−IR=BLv−IR= 0 (564)