Establish the expression for equivalent resistance of series comb… - Vidyadip

Question

Establish the expression for equivalent resistance of series combination and parallel combination of resistance separately.

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Answer

(a) Resistances in series combination : "A number of resistors are said to be connected in series combination if they are connected end to end."
In this combination when some potential difference is applied across the ends of the combination the same current flows through each of the resistors. But potential difference across each of the resistors is different and the sum of the potential difference across the individual resistors is equal to the potential difference applied across the combination.

In fig. (a) the series combination of three resistors having resistances $R _1, \quad R _2$ and $R _3$ is shown, when a potential difference V is applied across the ends A and B of the combination by connecting a cell between these points, let the current flowing through all the resistors be I. Suppose that the potential difference across the individual resistors of resistance $R _1, \ R _2$ and $R _3$ are $V _1, \ V _2$ and $V _3$ respectively. Then, according to Ohm's law,
$V _1= IR _1, V_2= IR _2$ and $V _3= IR _3$
Let the equivalent resistance of the combination be R, then according to Ohm's law,
V = IR.
But for series combination :
$V=V_1+V_2+V_3$
$\Rightarrow \quad IR = IR _1+ IR _2+ IR _3$
$\Rightarrow \quad R = R _1+ R _2+ R _3$
The same argument may be applied to the series combination of any number of resistors. Hence
$R = R _1+ R _2+ R _3+\ldots \ldots+ R _i \Rightarrow R =\sum_{i=1}^n R _i$
"Thus, equivalent resistance of series combination of any number of resistors is equal to the sum of their individual resistances and is always greater than the value of resistance of individual resistors."

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