Question

With the help of labelled circuit diagram derive a formula to find combined resistance (R) when two or more resistance $(R_1,R_2,R_3.....)$ are connected in parallel taking symbols potential difference $\left(V\right)$ and current $\left(I\right)$.

Answer 1

Consider two resistors connected in parallel, as shown in figure. It is clear, from the figure, that the potential drop 'V' across the two resistors is the same. In general, however, the currents I1 and I2 which flow through resistors R1 and R2, respectively, are different. According to Ohm's law, the equivalent resistance Req between B and A is the ratio of the potential drop V across these points and the current I which flows between them. This current must equal the sum of the currents I1 and I2 flowing through the two resistors, otherwise charge would build up at one or both of the junctions in the circuit.

Thus,

I=I1+I2
It follows that
1/Req= I/V= (I1+I2)/V= I1/V +I2/V
giving,
1/Req=1/R1 + 1/R2
Here, we have made use of the fact that the potential drop V is common to all three resistors. Clearly, the rule is
'The reciprocal of the equivalent resistance of two resistances connected in parallel is the sum of the reciprocals of the individual resistances. '