Question

Make a rule: how would you find the resistance of a parallel circuit with n identical resistors?

Answer 1

When n resistors are connected in parallel, it means they are connected to the same potential difference V:
$V=V_1 =V_2 =...=V_n$ (2)
It also means that the total current in the circuit is given by the sum of the currents flowing through each branch (each resistor) of the circuit:
$I=I_1+I_2 +...+I_n$ (1)

By using Ohm's law:
$I= \frac{V}{R}$
we can rewrite (1) as
$\frac{V}{R_{eq}} = \frac{V_1}{R_1}+ \frac{V_2}{R_2}+...+ \frac{V_n}{R_n}$
However, we said that the potential difference across each resistor is equal (eq.(2)), so we can rewrite the last formula as
$\frac{V}{R_{eq}} = \frac{V}{R_1}+ \frac{V}{R_2}+...+ \frac{V}{R_n}$
From which we find an expression for the equivalent resistance of n resistors in parallel:
$\frac{1}{R_{eq}}= \frac{1}{R_1}+ \frac{1}{R_2}+....+ \frac{1}{R_n}$