Simple Nature - Light and Matter

ad/Power in a resistor: the

rate at which electrical energy is

being converted into heat.

radiating light.^7 Since the circuit oscillates at a frequency^8 of 60 Hz,

the temperature doesn’t really have time to cycle up or down very

much over the 1/60 s period of the oscillation, and we don’t notice

any significant variation in the brightness of the light, even with a

short-exposure photograph.

Thus, what we really want to know is the average power, “aver-

age” meaning the average over one full cycle. Since we’re covering

a whole cycle with our average, it doesn’t matter what phase we

assume. Let’s use a cosine. The total amount of energy transferred

over one cycle is

E=

∫

dE

=

∫T

0

dE

dt

dt,

whereT= 2π/ωis the period.

E=

∫T

0

Pdt

=

∫T

0

Pdt

=

∫T

0

IoVocos^2 ωtdt

=IoVo

∫T

0

cos^2 ωtdt

=IoVo

∫T

0

1

2

(1 + cos 2ωt) dt

The reason for using the trig identity cos^2 x = (1 + cos 2x)/2 in
the last step is that it lets us get the answer without doing a hard
integral. Over the course of one full cycle, the quantity cos 2ωtgoes
positive, negative, positive, and negative again, so the integral of it
is zero. We then have

E=IoVo

∫T

0

1

2

dt

=

IoVoT

2

(^7) To many people, the word “radiation” implies nuclear contamination. Ac-
tually, the word simply means something that “radiates” outward. Natural
sunlight is “radiation.” So is the light from a lightbulb, or the infrared light
being emitted by your skin right now.
(^8) Note that this time “frequency” meansf, notω! Physicists and engineers
generally useωbecause it simplifies the equations, but electricians and techni-
cians always usef. The 60 Hz frequency is for the U.S.
Section 10.5 LRC circuits 633