W9_parallel_resonance.eps

286 Week 8: Faraday’s Law and Induction

Note that I stuck delta’s in so that I could relate the amount of energy per amount of volume or
energy density in the magnetic fieldto help us make theansatz^76 :

ηm=dUm

dV

=B

2

2 μ 0

(626)

which strangely matches our similar equation (deduced from very similar considerations for the
energy density in theelectricfield:

ηe=

dUe

dV

=

1

2

ǫ 0 E^2 (627)

There is something really sort of spooky about this – it is redolent^77 of as-yet undiscovered relation-
ships between the electric and magnetic fields. Soon, my child, soon we will understand this and a
great burst ofilluminationwill occur. Literally.

As was the case for capacitors, it isn’t enough to just make theansatz. We need to verify that
it works for at least one other geometry of inductor, ideally one witha varying field and inductance
we can compute. Our only real choice here is the toroidal solenoid.

Example 8.9.1: Energy in a Toroidal Solenoid

Suppose you have the very toroidal solenoid we study above, carrying a currentI. We can use
Ampere’s Law to find the magnetic field strengthB(r) inside the solenoid, of course. We can then
use it to find:
dU
dV

=B(r)

2

2 μ 0

(628)

if we multiply this out:

dU=

B(r)^2

2 μ 0

dV (629)

andintegrate both sides, we should getUm, the total energy stored in the magnetic field (according
to our ansatz).

Show that this is exactly equal to:

U=^1

2

LI^2 (630)

using theLyou found above.

Note that I’m not actually doing this for you, but I will help you one teensy bit. the volume
elementdVyou should use is the one of thicknessdrat radiusrwith heighth, or

dV= 2πrhdr (631)

Give it a shot, for homework. You can do it!

8.10: Eddy Currents

We have seen up above that a current loop resists being pulled fromorpushed into a magnetic field
because the field induces currents that exert forces that act against any change in flux. Just as this
is true for actual e.g. loops of wire, it is also true forbulk conductors! Any conducting material
such as a sheet of copper will resist being pushed into or pulled out ofa magnetic field, because the
changing field causes currents to loop through the entire conductor as if it were many, many parallel
wires. We call these currents “eddy currents”.

(^76) Physicsspeak for “inspired guess”...
(^77) Politespeak for “it stinks”...