W9_parallel_resonance.eps

180 Week 5: Resistance

V

V = I R

V

2

V = Q/C

2

1

3 3

4

V = I R (^55)
I
I 2 I 3 I 4
start
I 1
(a) (b)
Figure 57: (a) A single “generic” circuit loop; (b) A single “generic” circuit junction.

Kirchhoff’s Loop Rule

Consider the genericcircuit loopin figure 57 (a) above. The particular devices in this loop are not
too important – I drew a fairly arbitrary mix of the three devices we are aware of so far, but later
we will learn about still more devices we might want to put into a circuit to do some startlingly
useful things.

Let us imagine that we watch a charge +qmoving around this circuit loop in the direction of
the current beginning at the (arbtrary) point “start”. As it goesacross each potentialV 1 , V 2 , ...the
energy of the charge goes up, goes down, goes up, goes down. Bythe time it gets back to the start
position, its potential energy has changed by:

∆U=qV 1 +qV 2 +qV 3 +qV 4 +qV 5 =q

∑

i

Vi (342)

If∆U 6 = 0, then the charge gets back to its starting point with adifferent energythan the one it
started with! Its kinetic energy will have changed!

However this isalmostimpossible. Electrons in particular, as fermions, are nearlycompletely
incompressiblein a wire. This means that the current in any line segment is the same atall points
in the segment. Changes in the electric field thatproducesthe current at all points in the conductor
propagate nearlyinstantaneouslythroughout the entire loop, because the speed of light is very
large compared to the size of the loop. As potentials across the elements in the circuit vary, the
current adjusts almost instantaneously. Consequently within averytiny margin associated with this
propagation time, the net energy gain or loss of a charge in a pass around the circuit loop must be
zero!

This means that:

loop∑

i

Vi= 0 (343)

is a simple statement ofenergy conservationfor the charges as they progress around the loop. This
equation is known asKirchhoff’s Loop Rule, and we will use it repeatedly to write down equations
that lead to equations of motion for dynamical circuit loops or conditions that must be satisfied for
loops that carry steady state currents.

Kirchhoff’s Junction Rule

Consider the genericcircuit junctionin figure 57 (b) above. Again it doesn’t matter much what
devices are on any of the legs. Charge is conserved – it is neither created nor destroyed. The