Obtain the expression for the equivalent resistance for 3 resistors connected in parallel and also write the expression of equivalent resistance for connection of' `n' resistors.

Explanation

`rArr " " R_(1) ,R_(2) and R_(3)` are connected between a and b points as shown in figure.

`rArr` By connecting terminals of battery of voltage V with a and b, I current will be passed through `R_(1) , R_(2), R_(3) " are " I_(1), I_(2) , I_(3)` respectively.
`rArr` According to Ohm.s law, p.d. across `R_(1), R_(2), R_(3)` is V.
`rArr V = I_(1) R_(I) rArr I_(1) = (V)/(R_(1)) " " `....(1)
`V = I_(2) R_(2) rArr I_(2) = (V)/(R_(2)) " " ` ....(2)
and V = `I_(3) R_(3) rArr I_(3) = (V)/(R_(3)) ` ... (3)
`rArr` At point .a. ,
`I = I_(1) + I_(2) + I_(3) " "` ... (4)
By substituting values of equation (1), (2) and (3) in equation (4),
`I = (V)/(R_(1)) + (V)/(R_(2)) + (V)/(R_(3))`
`therefore (I)/(V) = (1)/(R_(1)) + (1)/(R_(2)) + (1)/(R_(3)) " " `[ Dividing by V]
`rArr " If" (I)/(V) = (1)/(R_(eq)) , ` then
`(1)/(R_(eq) ) = (1)/(R_(1)) + (1)/(R_(2)) + (1)/(R_(3)) `
If `R_(eq) ` is represented as `R_(p)`, then `(1)/(R_(p)) = (1)/(R_(1)) + (1)/(R_(2)) + (1)/(R_(3))`
`rArr` For equivalent resistance of .n. unequal resistors in parallel,
`(1)/(R_(eq)) = (1)/(R_(1)) + (1)/(R_(2)) + (1)/(R_(3)) + ...+ (1)/(R_(n))`
`rArr` For equivalent resistance of .n. equal resistors of resistance R in parallel,
`(1)/(R_(eq)) = (n)/(R)`
`therefore R_(eq) = (R)/(n)`