W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 265

The rate at which work is done on the rod is:

PF=F·v=−BLI·v=−

B^2 L^2 v 02

R

exp(

(

−

2 B^2 L^2 t

mR

)

(579)

which isexactly the samebut which has, of course, the opposite sign becauseFis slowing the rod
down! If we add the two, we see that:
PR+PF= 0 (580)

and energy is indeed conserved. The kinetic energy removed from the rod by the induced force
appears in the resistor as heat, precisely. Our “non-conservative” loop integral of the field is, in fact,
conservative after all!

At this point we know pretty much everything about this loop (we could easily findx(t), for
example, by integrating

∫

v(t)dt) and it all works out perfectly consistently. If nothing else, the
physics of the rod sliding in the magnetic field worksas if an electric field is induced around the
conducting loop which does indeed do work on the system that transforms its initial kinetic energy
into heat energy in the resistor as it slows down the sliding rod. Since the magnetic field itself is
incapable of doing work of this sort as it can only exert forces at right angles to the direction of
motion of a charged particle, we really have little choice but to believe that this electric field is
“real”, at least as real as the electric field we invented to describe the action-at-a-distance Coulomb
force so many weeks ago.

In the next section we will clearly state the conclusions of the first two chapters in the form of a
single equation: Faraday’s Law.

8.3: Faraday’s Law

In the last section, we saw that for the rod sliding down the rails (at least) we could describe
the voltage induced around the closed loop formed by the rails as thetime rate of change of the
magnetic flux through the loop. We left open the question of how to specify thedirectionof the
inducedE-field, although clearly we have to have just the right sign (direction) in order for energy
to be conserved as it was for the rod and resistor together.

If we point our right hand’s thumb in the direction of the magnetic fieldthrough the loop in
the previous section and let its fingers curl around the loop the natural direction to specify the
“positive” direction for the loop (clockwise as drawn in figure 92), then anincreasingloop area and
increasingflux produced anegativedirected electric field (counterclockwise as drawn) and induced
current that went the other way. This in turn made the force on the rodnegativeas it had to be,
it turned out, for energy to be correctly conserved. This suggests that we could have written the
voltage that appears in the loop completely consistently with respect to magnitude and direction
using this “right hand rule” as:

Vinduced in C=

∮

C

E~·d~ℓ=−d

dt

∫

S/C

B~·nˆdA (581)

This equation is known asFaraday’s Lawand is our first truly dynamical field equation for
the electromagnetic field. It tells us thatchanging magnetic fluxthrough an arbitrary loop creates
anelectric field around the loop. The minus sign on the right hand side tells us the direction of this
field – if we let the fingers of our right hand curling around the loop as our thumb points in the
(predominant) direction ofB~through the loop, then if the flux through the loop is increasing the
E-field circulates the loopCin the negative (right handed) direction; if the flux through the loopis
decreasing theE-field circulates aroundCin the positive direction.

The information encoded in this humble minus sign (which leads to energy conservation) is so
important that it has a name of its own – it is calledLenz’s Law. Lenz’s Law can be stated a
different way in words as well: