Page 366 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 29] ALTERNATING-CURRENT CIRCUITS 351

2
What is needed is the average value of I R over a complete cycle, which is
2
2
2
2
[I R] av = I R = I max R[sin ωt] av
eff
The average value of I over a complete cycle is zero because I is positive over half the cycle and negative over
2
the other half. However, I is always positive, and its average is therefore a positive quantity. As it happens, the
1
2
average value of sin ωt over a complete cycle is . Hence,
2
I max
2
1 2
I R = I R and so I eff = √
eff 2 eff = 0.707 I max
2
Similarly, the effective voltage in an ac circuit is
V max
V eff = √ = 0.707 V max
2
Currents and voltages in an ac circuit are usually expressed in terms of their effective values. The values of I
and V at any particular moment are called the instantaneous values of these quantities.
The effective values of I and V are sometimes called root-mean-square or rms values because of the way
they are derived.
SOLVED PROBLEM 29.1

Find the maximum voltage across a 120-V ac power line.

±V eff ±120 V
±V max = = =±170 V
0.707 0.707
The voltage across the power line varies between +170 V and −170 V.

SOLVED PROBLEM 29.2
The dielectric used in a certain capacitor breaks down at a potential difference of 300 V. Find the maxi-
mum effective ac potential difference that can be applied to it.

V eff = 0.707V max = (0.707)(300 V) = 212 V

SOLVED PROBLEM 29.3
Alternating current with a maximum value of 10 A is passed through a 20- resistor. At what rate does
the resistor dissipate energy?
The effective current is

I eff = 0.707I max = (0.707)(10 A) = 7.07 A
and so the power dissipated is
2
2
P = I R = (7.07 A) (20 ) = 1000 W
eff

REACTANCE
The inductive reactance of an inductor is a measure of its effectiveness in resisting the flow of an alternating
current by virtue of the self-induced back emf that the changing current causes in it. Unlike the case of a resistor,
there is no power dissipated in a pure inductor. The inductive reactance X L of an inductor whose inductance is