DC Circuit

# Combination of Resistors

Resistors are combined to increase or decrease total resistance in the circuit. Resistors can be combined in two ways; series and parallel.

## Resistors in Series

​In this type of combination, resistors are connected end to end consecutively. The combination of three resistors of resistances $R_1$, $R_2$ and $R_3$ in series is shown in figure given below;

​Consider a battery of voltage $V$ is connected to the ends of the combined resistors. Then, all the resistors have same current $I$ through them but the p.d. across each resistors will be different. Let $V_1$, $V_2$ and $V_3$ be the p.d. across the resistances $R_1$, $R_2$ and $R_3$ respectively. Then, $V=V_1+V_2+V_3$

From Ohm’s law, $V=IR_1+IR_2+IR_3$ $V=I(R_1+R_2+R_3)\text{ __(1)}$ Let $R$ be the resultant resistance of the circuit such that when p.d. $V$ is applied, current $I$ is through the circuit. $V=IR\text{ __(2)}$

From $(1)$ and $(2)$, $IR= I(R_1+R_2+R_3)$ $R= (R_1+R_2+R_3)$ For $n$ resistors in series, $R_{\text{eq}}=R_1+R_2+…+R_n$ Thus, in series combination, the equivalent resistance is equal to the sum of the individual resistances. And, the equivalent resistance is always greater than the individual resistance. So, to increase the resistance in the circuit, resistors are combined in series.

## Resistors in Parallel

​In this type of combination, resistors are connected between the same two points. The combination of three resistors of resistances $R_1$, $R_2$ and $R_3$ in parallel is shown in figure given below;

​Consider a battery of voltage $V$ is connected to the ends of the combined resistors and current $I$ is through the circuit. Then, all the resistors have same p.d. but the current $I$ is different in different resistors.

Let $I_1$, $I_2$ and $I_3$ be the current across the resistances $R_1$, $R_2$ and $R_3$ respectively. Then, $I=I_1+I_2+I_3$

From Ohm’s law, $I=\frac{V}{R_1}+\frac{V}{R_2}+\frac{V}{R_3}$ $I=V(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3})\text{ __(3)}$

Let $R$ be the resultant resistance of the circuit such that when p.d. $V$ is applied, current $I$ is through the circuit. $I=\frac{V}{R}\text{ __(4)}$ From $(3)$ and $(4)$, $\frac{V}{R}= V(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3})$ $\frac{1}{R}= \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$

For $n$ resistors in parallel, $\frac{1}{R_\text{eq}}= \frac{1}{R_1}+\frac{1}{R_2}+…+\frac{1}{R_n}$

Thus, in parallel combination, the reciprocal of the equivalent resistance is equal to the sum of the reciprocal of individual resistances. And, the equivalent resistance is always less than the individual resistance. So, to decrease the resistance in the circuit, resistors are combined in series.