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Current Electricity — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsCurrent Electricity4 Marks
Question
Establish the expression for equivalent parallel combination of resistance separately.
✓

Answer

Resistances in Parallel Combination : "A number of resistors are said to be connected in parallel combination when they are arranged in such a way that their first ends are joined at one point and the second ends at another point."
In this combination when some potential difference is applied across the ends of the combination, by connecting a cell between these points, then the potential difference between the ends of all the resistors is the same as that of applied potential difference but electric current in different resistors is different according to their resistance. The sum of individual currents is equal to the main current in the circuit.
In Fig. (b) three resistors of resistances $ R_1, R_2 $ and $ R_3 $ are connected in parallel combination between two points
Image
A and B. When a potential difference V is applied between these points by connecting a cell between the points, then let the current drawn from the cell in the circuit be I. The potential difference across the ends of individual resistors will also be V. At point A the current I is divided into three parts. Let $ I_1, I_2, $ and $ I_3 $ be the electric current in the resistors of the resistances $ R_1, R_2 $ and $ R_3 $, respectively. Let R be the equivalent resistance of this parallel combination.
According to Ohm's law,
$ V = IR \Rightarrow I = V/R $
and $ I_1 = V/R_1, I_2 = V/R_2 $ and $ I_3 = V/R_3 $
But in parallel combination:
$ I = I_1 + I_2 + I_3 $
$ \Rightarrow \frac{V}{R} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} $
$ \Rightarrow \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} $
"Thus the reciprocal of the equivalent resistance of the resistances of resistors connected in parallel is equal to the sum of the reciprocals of the individual resistances. The value of the equivalent resistance is always less than the value of the smallest resistance in the combination."

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