W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 267

magnetic flux through some surfaceSbounded by a closed curveC:

φm(t) =

∫

S/C

B~·ˆndA (583)

you will soon realize that the fluxφmcan vary in time for any or all offour reasons:

a)Ccan change in time (and hence so canS).

b) The magnitude ofB~can change in time.

c) The angle betweenB~andˆncan change in time because the direction ofB~changes.

d) The angle betweenB~andˆncan change in time because the direction ofˆnchanges.

Yes, one can imagine a loop that is changing its size and its orientation inside a magnetic field
that is changing its magnitude anditsorientation, all four changes in time contributing to the overall
change in magnetic flux through a surfaceSbounded by the loop! This multiplicity of ways the
magnetic flux depends on geometry and field strength makes it difficult to figure out the direction
of the induced field. In this section, we will endeavor to provide examples of each of theseseparately
to help you see how it all goes. With a bit of meditation, you should thenbe able to figure out how
to synthesize this knowledge and work out the direction when multiplethings are changing at once.

0.0.1 Lenz’s Law for changingC

B(in)

R

B(in)

R

I,E I,E

(a) (b)

Figure 94: Illustration ofE~-field direction for loops that change size. In (a) the loop is getting larger
(tending to increase the magnetic flux) so the induced magnetic moment from a counterclockwise
E~field and current opposes the existing field through the loop. In (b)the loop 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.

We’ve already seen an example of this in our single meaningful example this far. If a plane
loopCin a fixed magnetic field isincreasing in size, then the induced field points in the opposite
direction to the right handed direction determined from the magnetic field through the loops. If it
is decreasing, it points around the loopCin thesameright handed sense.

In terms of the verbal statement (illustrated in figure 94), if a conductor of resistanceRwere
placed along a pathCincreasingin area (in (a)), the current in the loop thus formed would have
a magnetic moment thatopposes the increasing fluxthrough the loop. Incidentally, the magnetic
force acting on this current would pointintowards the center of the loop which is the direction that
makes the loop try toshrink, not grow, opposing again the increase in flux.