Fundamental Engineering Concepts - Mathematics and Statistics for the Environment - page 275

I 1 I 2 I 3

I T

+

E 3

E 1 E 2

E R 1 R 2

R 3

-

FIGURE 11.41 Parallel circuit.

Note: Ohm's law states that the current in a circuit is inversely proportional to the circuit resis-
tance . This fact, important as a basic building block of electrical theory, is also important in
the following explanation of current flow in parallel circuits.

The behavior of current in a parallel circuit is best illustrated by example (see Figure 11.41).
The resistors R 1 , R 2 , and R 3 are in parallel with each other and with the battery. Each parallel path
is then a branch with its own individual current. When the total current ( I T ) leaves the voltage
source ( E ), part I 1 of current I T will flow through R 1 , part I 2 will flow through R 2 , and I 3 through
R 3 . The branch currents I 1 , I 2 , and I 3 can be different; however, if a voltmeter (used for measuring
the voltage of a circuit) is connected across R 1 , R 2 , and R 3 , then the respective voltages E 1 , E 2 , and
E 3 will be equal. Therefore,

E = E 1 = E 2 = E 3 (11.32)

The total current I T is equal to the sum of all branch currents:

I T = I 1 = I 2 = I 3 (11.33)

This formula applies for any number of parallel branches, whether the resistances are equal or
unequal.
By Ohm's law, each branch current equals the applied voltage divided by the resistance between
the two points where the voltage is applied. Hence, for each branch we have the following equations:

E
R

V
R

1
1

Branch 1: I ==

1

1

E
R

V
R

2
2

Branch 2: I == (11.34)

2

2

E
R

V
R

3
3

Bra nch3: I ==

3

3

With the same applied voltage, any branch that has less resistance allows more current through it
than a branch with higher resistance.

■ EXAMPLE 11.34
Problem: Two resistors, each drawing 2 amps, and a third resistor drawing 1 amp are connected in
parallel across a 100-V line (see Figure 11.42). What is the total current?