W9_parallel_resonance.eps

278 Week 8: Faraday’s Law and Induction

transformers and inductively coupled rectifiers and the like. Thereare some places where one can
make very clever use of mutual induction to accomplish some astounding things, such as in aTesla
Coil^74

8.7: Self-Induction

Now we get to one of the most important parts of this chapter: computing the self-inductance
of various simple current loops. We will have even fewer cases of geometries (and idealizations!)
where we can eventhinkof doing the integrals in a course at this level, and I will pretty much
present all of them here. Interested students can, and should,visit wikipedia here: Wikipedia:
http://www.wikipedia.org/wiki/Inductance both to read more aboutinductance itself and to see its
lovely table of the self-inductance of a number of circuit shapes withlessidealization. Nevertheless,
our idealized answers herein will be more than sufficient to help us fully understand both the essential
concepts and the general algebra required to do a better job.

Our general solution strategy here will be:

a) Find the magnetic field produced by the currentIin the loop in question. Usually we will use

Ampere’s Law for this simply because integrating the Biot-Savart Law for arbitrary points in

space is usually too difficult.

b) Write an expression for the flux produced by that field through the loop(s) that produce(s)

it. This may be a simple product of field times area (for constant field perpendicular to the

surface bounded by the loop) or an integral not unlike the one we didfor rectangular loops

near a long straight wire.

c) In cases where there are many “turns” (loops of wire) contributing to the overall flux, multiply

byN, the number of turns.

d) Divide out the current. Voila! The self-inductanceL!

Let’s start with the simplest and most important example, the moralequivalent of the parallel
plate capacitor for magnetic fields. The Self-Inductance of the (ideal) Solenoid:

Example 8.7.1: The Self-Inductance of the Solenoid

In figure 102 I’ve drawn an “ideal” circular cross-section solenoid, one withN(tightly wound) turns,
a radiusR, and a lengthℓ≫R. Obviously I’ve had to exaggerate some of these features in the
drawing – the radius of the wire itself is really very small compared to the other length dimensions,
there is very little space between turns, and it should be longer compared to its illustrative radius.

Following the rubric given above, we first find the field inside of the solenoid using Ampere’s Law
(see week 7 if you cannot remember the correct Amperian path to use as the curveC):
∮

C

B~·d~l = μ 0 Ithru C

Bb = μ 0 N

ℓ

Ib

B = μ 0

N

ℓ

I (605)

(^74) Wikipedia: http://www.wikipedia.org/wiki/Tesla Coil. ATesla Coil is basically a big resonant transformer that
makes Big Sparks. In fact, it pretty much makeslightning. As such, it is a great favorite for students to make for an
extra-credit project, because taming the lightning is whatphysics is all about, isn’t it...?