With a neat labeled diagram, derive the equation for three resistance connected in parallel.

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Solution

The above figure shows the circuit diagram for a parallel combination of three resistors with resistance $R_{1}, R_{2}, R_{3}$ connected between the points C and D. The Voltmeter measures the potential difference V between C and D and the ammeter measure the current in a given branch.
Let $I$ be the total current passing through the circuit. $I_{1}, I_{2}$ and $I_{3}$ be the current passing through the resistor of resistance $R_{1}, R_{2}$ and $R_{3}$ respectively.
$I = I_{1} + I_{2} + I_{3}$
If $R_{p}$ be resultant resistance of parallel combination, we have by Ohm's Law.
$I = \frac{V}{R_{p}}$ and $I_{1} = \frac{V}{R_{1}}, I_{2} = \frac{V}{R_{2}}$ and $I_{3} = \frac{V}{R_{3}}$ .
from equation (i)
$I = I_{1} + I_{2} + I_{3} = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}}$
$∴ I = V (\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}})$
$∴ \frac{V}{R_{p}} = V (\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}})$
$∴ \frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$

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