Week 8: Faraday’s Law and Induction 277
- From Ampere’s Law (show!):
B 1 =
μ 0 I 1
2 πr(598)
into the paper on the side of the loop.- To find the flux through the obvious plane surfaceSbounded by the rectangle, we have to
start by finding the flux in the differentially thin strip shaded in the figure. The magnetic
field is known and approximately constant in the strip in the limit that it isdifferentially thin.
Thus:
dφ 2 =
(
μ 0 I 1
2 πr)
adr (599)andφ 2 =(
μ 0 I 1 a
2 π)∫d+bddr
r=(
μ 0 I 1 a
2 π)
ln(
d+b
d)
(600)
- We can find the mutual inductance by dividing the flux byI 1 :
M 21 =
φ 2
I 1=
(μ 0 a
2 π)
ln(
d+b
d)
(601)
(This doesn’t really help us find the force, but it is certainly somethingyou should be able to
do.)- From Faraday’s Law (show!)
V 2 =−
dφ 2
dt=−
(μ 0 a
2 π)
ln(
d+b
d)
dI 1
dt(602)
and sinceI 1 isincreasing, we expect the voltage to decrease (and drive a current)counter-
clockwisefrom Lenz’s Law (see above).- From Kirchoff’s Rule and Ohm’s Law (show!)
V 2 −I 2 R= 0 (603)or
I 2 (t) =(μ 0 a
2 πR)
ln(
d+b
d)
dI 1
dt(604)
(counterclockwise fordIdt^1 >0).- Finally, the force on each wire is – naaaah, I’m too lazy to help you out any more. Besides, I
think you already found it in a previous homework assignment. The force on the side wires is
a bit tricky, mind you, but notthattricky and the final answer is now very simple to obtain.
What direction does the net force have to point evenbeforeyou work it out?
As noted, this is pretty much your first homework problem, given down below. While it is OK
to skim this part of the chapter before starting it, once you startitdo not look backat this example;
try very hard to work through the reason on your own. This means, of course (if you are reading
these words right before you start the homework, maybe you’d better skim through this example
aqainbefore you start...
There are a few other examples of “simple” geometries where one can compute the mutual induc-
tance, and you will do at least one other one on your homework. Theplace where mutual inductance
is acritical feature, the wholepointinstead of an annoyance is in the design and construction of