Deduce the expression for the the equivalent resistance of three resistors connected in series.

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Solution

Series combination of resistors : If a number of resistors are joined end to end so that the same current flows through each of them in succession, then the resistors are said to be connected in series.
As shown in the figure, consider three resistors $R_{1}, R_{2}, R_{3}$ connected in series. Suppose a current $I$ flows through the circuit when a cell of $V$ volt is connected across the combination.
By Ohm’s law, the potential differences across the three resistors will be,
$V_{1} = I R_{1}, V_{2} = I R_{2}, V_{3} = I R_{3}$
If $R_{s}$ be the equivalent resistance of the series combination, then on applying a potential difference $V$ across it, the same current $I$ must flow through it.
Therefore,
$V = I R_{s}$
But $V = V_{1} + V_{2} + V_{3} ∴, I R_{s} = I R_{1} + I R_{2} + I R_{3} ⟹ I R_{s} = I (R_{1} + R_{2} + R_{3}) ⟹ R_{s} = R_{1} + R_{2} + R_{3}$
Laws of resistances in series:
(i) Current through each resistance is same
(ii) Total voltage across the combination = sum of the voltage drops
(iii) Voltage drop across any resistor is proportional to its resistance
(iv) Equivalent resistance = sum of the individual resistances
(v) Equivalent resistance is larger than the largest individual resistance.

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