Obtain an expression for equivalent resistance of a series combination of three resistors.

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Solution

Take 3 resistors $r_{1}, r_{2}, r_{3}$ connected in parallel and connect a battery of $V$ volts.in this case the p.da cross the ends will be the same.the current flowing,however,is not the same.

total, $I = I_{1} + I_{2} + I_{3} ⟹ (1)$

from ohm's law:

$I = \frac{V}{r} ⟹ (2)$

$I_{1} = \frac{V}{r_{1}} ⟹ (3)$

$I_{2} = \frac{V}{r_{2}} ⟹ (4)$

$I_{3} = \frac{V}{r_{3}} ⟹ (5)$

putting the values of $I_{1}, I_{2}, I_{3} i n [1]$

$\frac{V}{r} = \frac{V}{r_{1}} + \frac{V}{r_{2}} + \frac{V}{r_{3}}$

$\frac{V}{r} = V [\frac{V}{r_{1}} + \frac{V}{r_{2}} + \frac{V}{r_{3}}]$

$\frac{V}{r} = \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}}$

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