W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 269

θ θ

n B n B

I,E I,E

Figure 96: Illustration ofE~-field direction when the direction ofB~orthe direction of the normal
to the loopnˆchanges. In (a) cos(θ) is getting larger (tending to increase the magnetic flux) so the
induced magnetic moment from a counterclockwiseE~field and current opposes the existing field
through the loop. In (b) cos(θ) is getting smaller (tending to decrease the flux) so the induced
magnetic moment from the clockwiseE~field and current supports the existing field through the
loop.

B~. If they are rotatingoutof alignment as shown in (b), cos(θ) is getting more negative and the
flux is decreasing, so the induced moment will support theB-field, resulting in a counterclockwise
current viewed from above the loop.

Note that it is entirely possible for all four of these contributions tothe total flux to be changing
at once. The loop and field could both be rotating, the loop could be shrinking or growing, and
the field could be turning on or turning offall at the same time! Problems where all of this is
going on at once are a bit excessive, perhaps, largely because it is such a pain to specify all of the
possibly competing parameters, butin principleyou know what you need to know to determine
theE~-field/current direction from Lenz’s Law. It will always point in the direction such that a
magnetic moment associated with a current in the inducedE~-field direction (whether or not one
actually exists) wouldoppose the change in magnetic flux through the loop.

That is precisely the right direction for energy conservation to always hold for the system. We
can breathe a sigh of relief!

Example 8.4.1: Wire and Rectangular Loop – Direction Only

R

b

a

d

I

r

B (in)

dr

Figure 97: A long straight wire sits next to a rectangular loop of wire and carries a currentIupas
shown. The current in the long straight wire can be increased or decreased.