270 Week 8: Faraday’s Law and Induction
In figure 97 above, a long straight wire is carrying a currentI. It sits a distancedaway from a
rectangular loop with side lengths ofaandb(all wires in the plane of the page) as shown.Ican be
increased or decreased at will.
Here’s the physics of this picture. The currentIcreates a magnetic field through the loop. We
can easily compute that field using Ampere’s Law (so we don’t have to remember things like the
magnetic field of long straight wires). On the other hand, we’ve worked enough with the magnetic
field of long straight wires that perhaps you do remember that it isμ 2 πr^0 Iinto the page for a current
up – I’ve helped you out a bit with lots of “dressing” on this figure thaton a quiz or exam you’d
have to provide for yourself.
IfIis varied, the field it generates varies as well. This changes the magnetic flux through the
rectangular loop. Mr. Faraday then tells us that there must be a voltage induced in the loop that
will create a current!
You can actuallycompletely calculatethe induced voltage in the rectangular loop using Faraday’s
Law (and will, in a homework problem) and from the voltage compute the current in the loop, and
from the current the force on the loop. But here our goal is more humble. We simply want to figure
out thedirectionof the induced current, and thedirectionof the induced force, using Lenz’s Law.
Suppose the currentIisincreasing. Then we expect the magnetic field into the page – and
the magnetic flux through the loop – to be increasing as well, and we can tell the following (highly
anthropomorphized) story:
The increasing flux makes the loopsad, because it is a very conservative loop. It hates change,
and is happy with things just the way that they are. It says to itself“Gosh, I’d really rather the
magnetic flux through menotchange, what can I do?” It then has the brilliant idea: Create an
electric field to drive a current around itself so that its own magneticmoment opposes the change in
flux! Perhaps it won’t keep the flux from changing altogether, but itwill ensure that the flux only
changesmore slowlythan it would without the induced current.
But which way is that? Well, a clockwise current would make the momentof the loop pointinto
the page, which would make the field through the loop even stronger, so that won’t work. Instead
the reactionary little loop makes the current counterclockwise. Now its own magnetic fieldopposes
the field due to the wire, and slows the rate of change of magnetic flux through itself. Eventually,
of course, the field might reach a new constant value as the current in the long straight wire stops
changing and the loop becomes happy again with no current at all.
The current in the counterclockwise direction has an additional bonus for the loop. It makes
the net force on the loop pointawayfrom the wire (as you can verify when you solve the problem
completely). If the loop is free to move, moving away from the wire moves it from a strong field
near the wire to a weaker field farther away from the wire! This, too, helps to keep the flux through
the loop from increasing, and is a part of the responses predicted by Lenz’s Law.
When you do this problem for homework, you will have to compute thenet magnetic flux through
the loop (in order to differentiate it to find the induced voltage). I’vehelped you out here by shading
a strip of lengthaand widthdr, a distancedrfrom the main wire. It should be pretty easy to
compute the fluxdφmthrough this strip, and then to sum up the total flux using integration between
suitable limits. Give it a try.