Derive expression for the total resistance of a circuit in which a few resistors are connected in parallel.

Explanation

Resistors in parallel combination : Resistors are said to be connected in parallel combination when one end of all the resistors is connected to one common terminal (say A) and other end of all resistors is connected to another common terminal (say B) and potential difference across, each resistor is same.
Let V = Potential difference applied by the battery
I = Total current produced by the battery

Current I gets divided into three parts `I_1 , I_2` and `I_3` flowing through resistors `R_1, R_2 and R_3` respectively. The p.d. across each resistor will be same, let it be V.
Here `I = I_1 + I_2 + I_3" " ... (1)`
Using Ohm.s law for 1st, 2nd and 3rd resistors,
we get `I_1 = V/R_(1) , I_(2) = V/R_2 and I_3 = V/R_3 and I = V/R_p`
Putting these values in eqn. (1), we get
`I=V/R_1+V/R_2+V/R_3`
`V/R_p=V(1/R_1+1/R_2+1/R_3)`
or `1/R_p=1/R_1+1/R_2+1/R_3`
where `R_p` = Effective or equivalent resistance of parallel combination of resistors.
Thus, the reciprocal of equivalent resistance of parallel combination is equal to the sum of the reciprocals of the individual resistances.