# Resistors In Series And Parallel Combinations

## Introduction

A circuit is composed of conductors (wire), power source, load, resistor, and switch. A circuit starts and ends at the same point. Usually, copper wire without insulation is used as a conductor. A switch is used to make or break a circuit. Resistors control the flow of the electric current in a circuit. A resistor is a passive element which means that it only consumes power but does not generate power. A load in a circuit consumes electrical energy and converts it into other forms of energy like light, heat, etc. A load can be a light bulb, fan, etc.

In an electric circuit, the different components can be connected either in series or in parallel to produce different resistive networks. Sometimes, in the same circuit, resistors can be connected in both parallel and series, across different loops to produce a more complex resistive network. These circuits are known as mixed resistor circuits. In the end, however, the total resistance should be known. It is important to know how to do this because resistors never exist in isolation. They are always part of a larger circuit that will have many resistors connected in different combinations. So how do we calculate this total resistance for resistors in series and parallel circuits? In this article, let us look at how the total resistance is calculated in a circuit with different resistor combinations.

## Resistors in Series

A circuit is said to be connected in series when the same amount of current flows through the resistors. In such circuits, the voltage across each resistor is different. In a series connection, if any resistor is broken or a fault occurs, then the entire circuit is turned off. The construction of a series circuit is simpler compared to a parallel circuit.

If the resistors in a circuit are all connected in series, like the figure given above, then the total resistance of the system is given by the following relation.

Rtotal = R1 + R2 + ….. + Rn

The total resistance of the system is just the sum total of individual resistances.
For example, consider the following sample problem.

A resistor having an electrical resistance value of 100 ohms, is connected to another resistor with a resistance value of 200 ohms. The two resistances are connected in series. What is the total resistance across the system?

Here, R1 = 100 Ω and R2= 200 Ω

Rtotal = 100 + 200 = 300 Ω

## Resistors in Parallel

A circuit is said to be connected in parallel when the voltage is same across the resistors. In such circuits, the current is branched out and recombines when branches meet at a common point. A resistor or any other component can be connected or disconnected easily without affecting other elements in a parallel circuit.

The figure above shows ‘n’ number of resistors connected in parallel. The total resistance here is given by the following relation.

$\frac{1}{R_{total}}$ = $\frac{1}{R_1}~ +~ \frac{1}{R_2} ~+~ ….~ +~ \frac{1}{R_n}$

Here, the reciprocal of the total resistance of the system is the sum of the reciprocals of resistances of the individual resistor in the circuit.

For the problem given above, what if the resistors were connected in parallel instead of in series? What is the total resistance in that case?

$\frac{1}{R_{total}}$ = $\frac{1}{100}~ + ~\frac{1}{200}$

= $\frac{(200~ +~ 100)}{20000}$

= $\frac{300}{20000}$

Therefore,

$R_{total}$ = $\frac{20000}{300}$ = $66.67 ~Ω$

## Summary

• A circuit is composed of conductors (wire), power source, load, resistor and switch.
• Resistors control the flow of the electric current in a circuit.
• A circuit is said to be connected in series when the same amount of current flows through the resistors.
• The total resistance of a series circuit is given by the following relation.

Rtotal = R1 + R2 + ….. + Rn

• A circuit is said to be connected in parallel when the voltage is same across the resistors.
• The total resistance of a parallel circuit is given by the following relation.

$\frac{1}{R_{total}}$ = $\frac{1}{R_1}~ +~ \frac{1}{R_2} ~+~ ….~ +~ \frac{1}{R_n}$

• Sometimes, in the same circuit, resistors can be connected in both parallel and series, across different loops to produce a more complex resistive network.These circuits are known as mixed resistor circuits.

Learn to apply your conceptual knowledge to real-world applications of different combinations of resistors with us at Byju’s

#### Practise This Question

Find the reading of the spring balance shown in figure. The elevator is going up with an acceleration of g/10, the pulley and the string are light and the pulley is smooth.