W9_parallel_resonance.eps

276 Week 8: Faraday’s Law and Induction

Henries can, of course, also be expressed in terms of Webers – youdoremembers what Webers
are, don’t you? It should be fairly obvious that 1 Henry is 1 Weber/Second, but nobody cares much
about Webers, while everybody cares about Henries.

Example 8.6.1: The Mutual Inductance of a Wire and Rectangular Current Loop

R

b

a

d

r

B (in)

dr

I

I 2

1

Figure 101: A long straight wire carrying a time-varying currentIi(t) near a rectangular current
loop induces a voltageV 2 in that loop, which in turn creates a currentI 2 in that wire and a force
F~ 2 on the wire loop.

In figure 101 we return to the long straight wire and adjacent rectangular loop of wire (all in a
common plane) that we examined above in the limited context of Lenz’sLaw and the direction of
the induced current. This time, we want to answerallof the questions we might ask, such as:

• What is the magnetic field due toI 1 (Ampere’s Law, of course).


• Given this field, what is themagnetic fluxthrough the rectangular loopφ 2?


• Given this flux, what is themutual inductanceM 21?


• Given this flux and a currentI 1 (t) that isincreasing, what is the voltageV 2 induced in the
  rectangular loop?


• Given this voltage, what is the currentI 2 (t) in the rectangular loop (magnitude and “direction”,
  that is clockwise or counterclockwise in the arrangement shown?


• Finally, given this current, what is the netforceon the loop, and is it attractive (back towards
  the long straight wire) or repulsive?

That’s a lot of questions, but I laid it out in this way so you can see the very simple flow of
reason. In aquiz or examproblem I’d be much more likely to just give the picture (without any
“dressing”) and sayI 1 (t) =IT^0 t, what isF~ 2 (t)? So practice thinking about how this chain works
so that each answer is a trivial step away from the previous one, but put together the answer isn’t
“simple” at all!

At this point you should really all be able to answer each and every step on your own, so I’ll
provide the most cursory review of each step and let you fill in the details (completely, of course!)
for homework.