Name: Introduction to Electric Charge
Uploaded: 2019-07-19
Duration: 9 min 41 s
Description: In this video, we will study electric charge and its properties and we will also learn about different methods of charging.

Question

With the help of a circuit diagram derive a formula for the resultant resistance of three resistances connected in parallel.

Accepted Answer

A parallel circuit has two or more loops through which current can pass. If the circuit is disconnected in one of the loops, current can still pass through the other loop(s). The wiring in house consists of parallel circuits.

Let three resistors $R_{1}, R_{2}$ and $R_{3}$ are connected across two common points A and B. The potential difference across each resistance is the same and equal to the potential difference between A and B. This is measured using the voltmeter. The current after reaching to A, divides into three branches $I_{1}, I_{2}$ and $I_{3}$ passing through $R_{1}, R_{2}$ and $R_{3}$ respectively. According to the Ohm's law,

$I_{1} = \frac{V}{R_{1}}, I_{2} = \frac{V}{R_{2}}$ and $I_{3} = \frac{V}{R_{3}}$ .

The total current through the circuit is given by $I = I_{1} + I_{2} + I_{3}$ .

So $I = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}} . . . . . . . . . . (1)$

Let the effective resistance of the parallel combination of resistors be $R_{P}$ . Then

$I = \frac{V}{R_{P}} . . . . . . . . . . (2)$

On combining equations $(1)$ and $(2)$ ,

$\frac{V}{R_{P}} = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}}$

$\frac{1}{R_{P}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$

Thus, number of resistors are connected in parallel, the sum of the reciprocal of the individual resistances is equal to the reciprocal of the effective or equivalent resistance.

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