Explain the series connection of resistors and derive the formula of equivalent resistance.

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Solution

The resistors are connected across two points in the circuit in such a way that the current flowing through each resistor is the same and only one path is available for it to flow, then the resistors are said be connected in series.\In the figure above, three resistors $R_{1}, R_{2}$ and $R_{3}$ are connected in series across the points $A$ and $B$ . Hence, the current (I) flowing through each of the resistors is same but the total voltage of the battery is divided.If the voltage drops across $V_{1}, V_{2}$ and $V_{3}$ are $R_{1}, R_{2}$ and $R_{3}$ respectively then,
$V = V_{1} + V_{2} + V_{3}$ …………….(i)
Now, if the resistor R instead of these three resistances $R_{1}, R_{2}$ and $R_{3}$ is connected in such a way that the current flowing through the circuit remains the same, then $R$ is called equivalent resistance of the circuit.For an equivalent resistor, $V = I R$ …………….(ii)According to Ohm’s law,
$R_{1}$ voltage drops across, $V_{1} = I R_{1}$ …………..(iii)
$R_{2}$ voltage drops across, $V_{2} = I R_{2}$ …………..(iv)
$R_{3}$ voltage drops across, $V_{3} = I R_{3}$ …………..(v)
Adding (iii), (iv) and (v):
$V_{1} + V_{2} + V_{3} = I R_{1} + I R_{2} + I R_{3}$ ……(vi)Or,
$V = I R = I (R_{1} + R_{2} + R_{3})$ Or,
$R = R_{1} + R_{2} + R_{3}$
Thus, in series connection the equivalent resistor $R$ is equal to the sum of all the resistors connected in series.

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