Derive an expression for the effective resistance when three resistors are connected in (i) series (ii) parallel.

Explanation

Effective resistance when three resistors are connected:
(i) In series

(1) Three resistors `R_(1), R_(2)` and `R_(3)` are connected is series as shown in fig. `V_(1), V_(2), V_(3)` are the potential differences across `R_(1), R_(2)` and `R_(3)`. I is the current flowing through them.
(2) Applying Ohm's law to `R_(1), R_(2)` and `R_(3)`, Then `V_(1) = IR_(1), V_(2) = IR_(2), V_(3) = IR_(3)`
3) In series, `V = V_(1) + V_(2) + V_(3)`
`IR_(s) = IR_(1) + IR_(2) + IR_(3)` `[:' V = IR_(s)]`
`:. R_(s) = R_(1) + R_(2) + R_(3)`
in parallel

(1) Three resistors `R_(1), R_(2)` and `R_(3)` are connected in parallel as shown in fig. Potential differences across each resistor is `V. I_(1), I_(2), I_(3)` are the currents flowing through them.
(2) `V = I_(1)R_(1) = I_(2)R_(2) = I_(3)R_(3)`
`implies I_(1) = (V)/(R_(1)), I_(2) = (V)/(R_(1)), I_(3) = (V)/(R_(3))`
(3) In parallel, `I = I_(1) + I_(2) + I_(3)`
`implies (V)/(R_(p)) = (V)/(R_(1)) + (V)/(R_(2)) + (V)/(R_(3))` `[:' I = (V)/(R_(P))]`
`:. (1)/(R_(P)) = (1)/(R_(1)) + (1)/(R_(2)) + (1)/(R_(3))`