Combination of Resistors

Table of Contents

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 R₁, R₂ and R₃ in series is shown in figure given below;

Series Combination of Resistors

​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 V₁, V₂ and V₃ be the p.d. across the resistances R₁, R₂ and R₃ 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 appled, 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, \[R_{eq}=R_1+R_2+…+R_n\] 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 R₁, R₂ and R₃ in parallel is shown in figure given below;

Parallel Combination of Resistors

​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 I₁, I₂ and I₃ be the current across the resistances R₁, R₂ and R₃ 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 appled, 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, \[\frac{1}{R}= \frac{1}{R_1}+\frac{1}{R_2}+…+\frac{1}{R_n}\] 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.

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