Obtain an expression for equivalent resistance of two resistors connected in parallel.

Explanation

Consider two resistors of resistances `R_1 and R_2` connected in parallel is as shown.fig.
Let V be the potential difference across each resistor.
Let `I_(1) ,I_2` be the current through the resistors respectively.
Let I be the total current in the combination.
Let `R_e` be the effective resistance.
In parallel combination `I=I_(1)+I_(2) " "........(1)`
From Ohm.s law, V=IR or I =V/R to each resistor, we get
`I_(1) = V/ R_(1) and I_(2) = V/R^2`
`(1)implies I=(V)/R_(1)+V/R_(2)`
`= V[(1)/(R_(1)+1/R_(2))] " " ......(2)`
If the combination is replaced by effective resistor of resistance , `R_p` , we get
`I=V/R_(p) " "...(3)`
on comparing (2) and (3) we get
`V/R_(p)=V[1/R_1+1/R_2]`
`1/R_p=[1/R_1+1/R_2]" or " 1/R_p=(R_1+R_2)/(R_1R_2)`
or `R_(p) = (R_1R_(2))/(R_1+R_2)`
For .n. resistors in parallel combination
`1/R_(p) = [1/R_(1)+1/R_(2)+1/R_(3)+.........+1/R_(n)]`