W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 263

which is just begging to be turned into theflux of the magnetic field through the loopC:

|∆Vind|=

∮

C

E~ind·d~ℓ=dφm

dt

(571)

where:

φm=

∫

S/C

B~·nˆdA (572)

is themagnetic fluxthrough the surfaceSbounded by the closed loopC.

The second is that if energy isn’t ultimately conserved, life is going to bebadfor physics students
because magic^71 and perpetual motion machines both become possible, and yet we never seem to
actually observe either one in nature. Nature isstable, not unstable the way it would be if induced
forcesincreasedthe very motion that induced those forces (to make them increaseeven faster, with
no source for the energy associated with the ever-increasing force).

We’ve already seen that the potential around the loop has toincreasewhen we go counterclock-
wise in order to balance the rate that energy isremovedfrom the loop by the total resistance in
Kirchoff’s rule. Eventually we’re going to need to formalize this as a rulefor thesignof the change
in potential we get going around the loop in any given direction. In order for us to be able to tell
somebody far away about this rule, we ought to make sure that it is based on the use of our right
hands to determine loop directions relative to something that uniquely orients the problem, such as
the direction of the magnetic field through the loop.

Problem and Solution

In the next section we will, as promised, take all of these observations and combine them into a
new physical law, and a very beautiful one it will turn out to be! But yeah, let’s finish offthis
problem first. Of course you may be askingwhat problem, since I haven’t stated one yet. How’s this:
Let’s findeverythingabout this system, assuming only that it starts at timet= 0 moving at initial
velocityv 0 to the right.v(t),I(t), and so on, find it all. Time to use Newton’s Laws once again!

We begin with:

∮ ∆Vind−IR = 0

C

E~ind·d~ℓ−IR = 0

BLvx−IR = 0 (573)

(where we have used the results of the first section to evaluate the total induced voltage in the loop
and where we’ve added thexsubscript tovto make it clear that we are dealing only withx-directed
motion and force) or:

I=

BLvx

R (574)

in the direction (counterclockwise) shown around the loop.

Next, compute the force acting on the rod.Iflowsupperpendicular toB~whenvxis positive,
so Newton’s Second Law becomes:

Fx=−ILB=m

dvx

dt

(575)

(^71) You might, if you are a science fiction and fantasy reader (andwriter) like myself, think that it would be great
fun to live in a Universe where either one was possible. Thinkagain. Life is unstable, chaotic, and whimsical enough
as it is with thenegativefeedback associated with the laws of thermodynamics; with unbounded positive feedback
loops possible at all, it seems rather likely that the Universe would simply explode instantly, much the same way that
positive feedback in an amplifier leads to an ear-shatteringscreech and (if the gain is turned up enough) blown fuses.
We wouldn’t want to live in a Universe with a blown fuse now, would we?