3.10 COMBINATION OF RESISTORS – SERIES AND PARALLEL

The current through a single resistor \( R \) across which there is a potential difference \( V \) is given by Ohm’s law \( I = V/R \). Resistors are sometimes joined together and there are simple rules for calculation of equivalent resistance of such combination.

Two resistors are said to be in series if only one of their end points is joined (Fig. 3.13). If a third resistor is joined with the series combination of the two (Fig. 3.14), then all three are said to be in series. Clearly, we can extend this definition to series combination of any number of resistors.

Two or more resistors are said to be in parallel if one end of all the resistors is joined together and similarly the other ends joined together (Fig. 3.15).

Consider two resistors \( R_1 \) and \( R_2 \) in series. The charge which leaves \( R_1 \) must be entering \( R_2 \). Since current measures the rate of flow of charge, this means that the same current I flows through \( R_1 \) and \( R_2 \). By Ohm’s law: \[ \text{Potential difference across } R_1=V_1=IR_1 \text{, and} \] \[ \text{Potential difference across }R_2=V_2=IR_2. \] The potential difference \( V \) across the combination is \( V_1+V_2 \).

\[ V=V_1+V_2=I(R_1+R_2) \](3.36)

This is as if the combination had an equivalent resistance \( R_{eq} \), which by Ohm’s law is

\[ R_{eq}=\frac{V}{I}=I(R_1+R_2) \](3.37)

If we had three resistors connected in series, then similarly

\[ V=I R_{1}+I R_{2}+I R_{3}=I\left(R_{1}+R_{2}+R_{3}\right) \](3.38)

This obviously can be extended to a series combination of any number \( n \) of resistors \( R_1 \), \( R_2 \) ....., \( R_n \). The equivalent resistance \( R_{eq} \) is

\[ R_{eq}=R_1+R_2+...+R_n \](3.39)

Consider now the parallel combination of two resistors (Fig. 3.15). The charge that flows in at \( A \) from the left flows out partly through \( R_1 \) and partly through \( R_2 \). The currents \( I \), \( I_1 \), \( I_2 \) shown in the figure are the rates of flow of charge at the points indicated. Hence,

\[ I=I_1+I_2 \](3.40)

The potential difference between \( A \) and \( B \) is given by the Ohm’s law applied to \( R_1 \)

\[ V=I_1R_1 \](3.41)

Also, Ohm’s law applied to \( R_2 \) gives

\[ V=I_2R_2 \](3.42)

\[ \therefore I=I_{1}+I_{2}=\frac{V}{R_{1}}+\frac{V}{R_{2}}=V\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right) \](3.43)

If the combination was replaced by an equivalent resistance \( R_eq \), we would have, by Ohm’s law

\[ I=\frac{V}{R_{eq}} \](3.44)

Hence,

\[ \frac{1}{R_{eq}}=\frac{1}{R_1}+ \frac{1}{R_2} \](3.45)

We can easily see how this extends to three resistors in parallel (Fig. 3.16).

Exactly as before

\[ I=I_1+I_2+I_3 \](3.46)

and applying Ohm's law to \( R_1 \), \( R_2 \) and \( R_3 \) we get,

\[ V=I_{1} R_{1}, V=I_{2} R_{2}, V=I_{3} R_{3} \](3.47)

So that

\[ I=I_{1}+I_{2}+I_{3}=V\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\right) \](3.48)

An equivalent resistance \( R_{eq} \) that replaces the combination, would be such that

\[ I=\frac{V}{R_{eq}} \](3.49)

and hence

\[ \frac{1}{R_{e q}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}} \](3.50)

We can reason similarly for any number of resistors in parallel. The equivalent resistance of \( n \) resistors \( R_1 \), \( R_2 \) . . . ,\( R_n \) is

\[ \frac{1}{R_{e q}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\ldots+\frac{1}{R_{\mathrm{n}}} \](3.51)

These formulae for equivalent resistances can be used to find out currents and voltages in more complicated circuits. Consider for example, the circuit in Fig. (3.17), where there are three resistors \( R_1 \), \( R_2 \) and \( R_3 \). \( R_2 \) and \( R_3 \) are in parallel and hence we can replace them by an equivalent \( R_{eq}^{23} \) between point \( B \) and \( C \) with

\[ \begin{aligned} & \frac{1}{R_{e q}^{23}}=\frac{1}{R_{2}}+\frac{1}{R_{3}} \\ \text { or, } \mathrm{R}_{e q}^{23} &=\frac{R_{2} R_{3}}{R_{2}+R_{3}} \end{aligned} \](3.52)

The circuit now has \( R_1 \) and \( R_{eq}^{23} \) in series and hence their combination can be replaced by an equivalent resistance with

\[ R_{e q}^{123}=R_{e q}^{23}+R_{1} \](3.53)

If the voltage between \( A \) and \( C \) is \( V \), the current \( I \) is given by

\[ \begin{aligned} I &=\frac{V}{R_{e q}^{123}}=\frac{V}{R_{1}+\left[R_{2} R_{3} /\left(R_{2}+R_{3}\right)\right]} \\ &=\frac{V\left(R_{2}+R_{3}\right)}{R_{1} R_{2}+R_{1} R_{3}+R_{2} R_{3}} \end{aligned} \](3.54)