Page 15 - Schaum's Outline of Theory and Problems of Electric Circuits

INTRODUCTION
[CHAP. 1
4
EXAMPLE 1.3. In an electric circuit an energy of 9.25 mJ is required to transport 0.5 mC from point a to point b.
What electric potential difference exists between the two points?
9:25 10 6 J
1 volt ¼ 1 joule per coulomb V ¼ ¼ 18:5V
0:5 10 6 C

1.5 ENERGY AND ELECTRICAL POWER
Electric energy in joules will be encountered in later chapters dealing with capacitance and induc-
tance whose respective electric and magnetic fields are capable of storing energy. The rate, in joules per
second, at which energy is transferred is electric power in watts. Furthermore, the product of voltage
and current yields the electric power, p ¼ vi;1 W ¼ 1V 1 A. Also, V A ¼ðJ=CÞ ðC=sÞ¼ J=s ¼ W.
In a more fundamental sense power is the time derivative p ¼ dw=dt, so that instantaneous power p is
generally a function of time. In the following chapters time average power P avg and a root-mean-square
(RMS) value for the case where voltage and current are sinusoidal will be developed.

EXAMPLE 1.4. A resistor has a potential difference of 50.0 V across its terminals and 120.0 C of charge per minute
passes a fixed point. Under these conditions at what rate is electric energy converted to heat?
ð120:0C=minÞ=ð60 s=minÞ¼ 2:0A P ¼ð2:0AÞð50:0VÞ¼ 100:0W
Since 1 W ¼ 1 J/s, the rate of energy conversion is one hundred joules per second.

1.6 CONSTANT AND VARIABLE FUNCTIONS

To distinguish between constant and time-varying quantities, capital letters are employed for the
constant quantity and lowercase for the variable quantity. For example, a constant current of 10
amperes is written I ¼ 10:0 A, while a 10-ampere time-variable current is written i ¼ 10:0 f ðtÞ A. Exam-
ples of common functions in circuit analysis are the sinusoidal function i ¼ 10:0 sin !t ðAÞ and the
exponential function v ¼ 15:0 e at (V).

Solved Problems

2
1.1 The force applied to an object moving in the x direction varies according to F ¼ 12=x (N).
(a) Find the work done in the interval 1 m x 3m. (b) What constant force acting over the
same interval would result in the same work?
ð 3 3
ðaÞ dW ¼ Fdx so W ¼ 12 dx ¼ 12 1 ¼ 8J
1 x 2 x 1
ðbÞ 8J ¼ F c ð2mÞ or F c ¼ 4N

1.2 Electrical energy is converted to heat at the rate of 7.56kJ/min in a resistor which has 270 C/min
passing through. What is the voltage difference across the resistor terminals?
From P ¼ VI,
3
P 7:56 10 J=min
V ¼ ¼ ¼ 28 J=C ¼ 28 V
I 270 C=min