W9_parallel_resonance.eps

170 Week 5: Resistance

We can also look ahead a bit at this point. Soon we will discover that Maxwell’s equations
are called Maxwell’s equations because Maxwell more or less discovered aninconsistency in the
treatment of current in the original form of one of the laws that could only be made consistent by
adding a term to it toaccountfor the implications of charge conservation and the arbitrariness of
the infinity of surfaces “through” which charge can flow that are all bounded by a single closed curve
C.

To help the interested or advanced student out, consider the following picture: In this figure, we

C

J Q

S 1 S^2

n

n

Figure 51: Charge Conservation.

split the closed surfaceSboundingVinto two pieces,S=S 1 +S 2 by drawing single closed curve
Call the way around it.S 1 on the left andS 2 on the right are ballooned out so that they resemble
two “fishing nets” (in analogy with a worked problem) placed face to face, through which charge
can flow as illustrated in figure 51.

If current is flowing in a “steady state” way andcharge is conserved, the current from left
to right through the two surfacesS 1 andS 2 must be equal – the current through the first must
equal the current through the second becausein the steady state, no charge is building up in
between.

However, if we put e.g. a capacitor plate in between the two surfaces (or charge is accumulating
in some other way), current may not be flowing in a steady state way– current may bebuilding up
inside theclosedsurfaceS. In that case thedifferencebetween the current throughS 1 and the
current throughtS 2 is the rate at which charge builds up insideV:
∫

S 1

J~·ˆndA−

∫

S 2

J~·ˆndA=Iinto V thruS 1 −Iout of V thruS 2 =d

dt

∫

V/S

ρedV=

dQinS/V

dt

(302)

Note that all we really did to get this result is split the integral over the closed surfaceSinto two
pieces, and change the sign/direction ofnˆfor the first surface so that they both go “left to right”
instead of “out”. You should verify that this makes sense on your own.

Armed with this result, students are encouraged to “play Maxwell” as they go along, and see if
they can discover and fix this inconsistencyall by themselveswithout looking ahead to see how it is
done when Ampere’s Law is introduced. You now have all the information you need to do so except
for, of course, the actual equation that needs to be repaired which is covered in a later chapter.
When you cover it, your instructor may point it out and suggest that you give it a try.

The Drude Model

The earliest forms of Wikipedia: http://www.wikipedia.org/wiki/pinball worked by dropping a small
metal sphere (usually a ball bearing of some sort) into a vertical box fronted by a piece of glass and
studded on the inside with “pins” as pictured in figure 52. The ball would bounce down through
the pins in not-quite-random ways and end up in one of several slotsat the bottom. One could then
gamble on just which box a ball would end up in, or try to use skill in the way the ball was dropped
to determine the outcome, but because the (essentially classical) motion was effectively chaotic, it