Resistance in Parallel

When resistors are placed in parallel, they are arranged side by side with their corresponding ends joined together at two common junctions. With the arrangement, the potential difference across the resistors is the same.

The positive terminals of all the resistors are connected to the same positive terminal and the negative terminals of the resistors are connected to the same negative terminal.

The equipment parallel resistance R_total is found using the expression,

I = I₁ + I₂ + I_{3 ____________________} (1)

Since the current flowing through each resistor is 

I₁ = \( \frac{V}{R_1}\), I₂ = \( \frac{V}{R_2}\), I₃ = \( \frac{V}{R_3}\)   __________________ (2)

Substituting these equations (2) in (1)

 ∴ I = \( \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} =\frac{V}{R_{total}}\)

= \( \scriptsize V \left ( \normalsize \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right) = \normalsize \frac{V}{R_{total}}\)

cross multiply

\(\frac{V}{V(R_{total})} = \frac{1}{R_1}+ \frac{1}{R_2} + \frac{1}{R_3}\)