Page 306 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 25] DIRECT-CURRENTCIRCUITS 291

A calculator with a reciprocal (1/X) key makes it easy to find the equivalent resistance of a set of resistors
in parallel. The key sequence would be

[R 1 ][1/X][+][R 2 ][1/X][+][R 3 ][1/X][+] ··· [=][1/X]

This method is much faster than working out the calculation one term at a time. What is being done here is to
replace the formula 1/R = 1/R 1 + 1/R 2 + 1/R 3 + ··· by its reciprocal

1
R =
1/R 1 + 1/R 2 + 1/R 3 +· · ·
If only two resistors are connected in parallel,

1 1 1 R 1 + R 2 R 1 R 2
= + = and so R =
R R 1 R 2 R 1 R 2 R 1 + R 2

SOLVED PROBLEM 25.6
Show that the equivalent resistance of three resistors in parallel is given by 1/R = 1/R 1 + 1/R 2 + 1/R 3 .
To find the equivalent resistance, we start from the fact that the total current I is equal to the sum of the currents
through the separate resistors:

I = I 1 + I 2 + I 3
Because the potential difference V is the same across all the resistors, their respective currents are
V V V
I 1 = I 2 = I 3 =
R 1 R 2 R 3
The smaller the resistance, the greater the current through a resistor in a parallel set. The total current is given in
terms of the equivalent resistance R by
V
I =
R
Substituting for the I’s in I = I 1 + I 2 + I 3 gives
V V V V
= + +
R R 1 R 2 R 3
Now we divide both sides of this equation by V :
1 1 1 1
= + +
R R 1 R 2 R 3

SOLVED PROBLEM 25.7
(a) What is the equivalent resistance of three 5- resistors connected in parallel? (b) If a potential
difference of 60 V is applied across the combination, what is the current in each resistor?
1 1 1 1 1 1 1 3
(a) = + + = + + =
R R 1 R 2 R 3 5 5 5 5
5
R = = 1.67
3
(b) Since each resistor has a potential difference of 60 V across it, the current in each one is
V 60 V
I = = = 12 A
R 5