Simple Nature - Light and Matter

z/In a capacitor, the current

is 90◦ ahead of the voltage in

phase.

aa/Representing functions

with points in polar coordinates.

ab/Adding two sinusoidal

functions.

tuner is hooked up to a receiving antenna. We know that a current
will flow in the circuit, and we know that there will be resonant
behavior, but it is not necessarily simple to relate current to voltage
in the most general case. Let’s start instead with the special cases
of LRC circuits consisting of only a resistance, only a capacitance,
or only an inductance. We are interested only in the steady-state
response.
The purely resistive case is easy. Ohm’s law gives

I=

V

R

.

In the purely capacitive case, the relationV =q/Clets us cal-

culate

I=

dq

dt

=C

dV

dt

.

This is partly analogous to Ohm’s law. For example, if we double

the amplitude of a sinusoidally varying AC voltage, the derivative

dV/dtwill also double, and the amplitude of the sinusoidally varying

current will also double. However, it is not true thatI=V/R, be-

cause taking the derivative of a sinusoidal function shifts its phase by

90 degrees. If the voltage varies as, for example,V(t) =Vosin(ωt),

then the current will beI(t) =ωCVocos(ωt). The amplitude of the

current isωCVo, which is proportional toVo, but it’s not true that

I(t) =V(t)/Rfor some constantR.

A second problem that crops up is that our entire analysis of

DC resistive circuits was built on the foundation of the loop rule

and the junction rule, both of which are statements about sums. To

apply the junction rule to an AC circuit, for exampe, we would say

that the sum of the sine waves describing the currents coming into

the junction is equal (at every moment in time) to the sum of the

sine waves going out. Now sinusoidal functions have a remarkable

property, which is that if you add two different sinusoidal functions

having the same frequency, the result is also a sinusoid with that

frequency. For example, cosωt+ sinωt=

√

2 sin(ωt+π/4), which

can be proved using trig identities. The trig identities can get very

cumbersome, however, and there is a much easier technique involv-

ing complex numbers.

Figure aa shows a useful way to visualize what’s going on. When

a circuit is oscillating at a frequencyω, we use points in the plane to

represent sinusoidal functions with various phases and amplitudes.

self-check I

Which of the following functions can be represented in this way? cos(6t−

4), cos^2 t, tant .Answer, p. 1060

Section 10.5 LRC circuits 629