Page 303 - Schaum's Outline of Theory and Problems of Applied Physics

CHAPTER 25

Direct-Current

Circuits

RESISTORS IN SERIES
The equivalent resistance of a set of resistors connected together is the value of the single resistor that can replace
the entire set without changing the properties of any circuit of which the set is a part. The equivalent resistance
of a set of resistors depends on the way in which they are connected as well as on their values. If the resistors
are joined in series, that is, consecutively (Fig. 25-1), the equivalent resistance R of the combination is the sum
of the individual resistances:

R = R 1 + R 2 + R 3 +· · · series resistors

Fig. 25-1

SOLVED PROBLEM 25.1
Show that the equivalent resistance of three resistors in series is given by R = R 1 + R 2 + R 3 .
To find the equivalent resistance, we start from the fact that the potential difference V across the set is the sum
of the potential differences across the individual resistors:

V = V 1 + V 2 + V 3
Because the current in each resistor is I, the potential differences across them are

V 1 = IR 1 V 2 = IR 2 V 3 = IR 3
The potential difference across the equivalent resistance R is
V = IR

Substituting for the V ’s in V = V 1 + V 2 + V 3 gives
IR = IR 1 + IR 2 + IR 3
Now we divide both sides of this equation by I and find that

R = R 1 + R 2 + R 3
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