Question

Derive the expression for the equivalent resistance of three resistors R₁, R₂ and R₃ connected in parallel.

Answer 1

The current I divides into three parts: I₁ through R₁, I₂ through R₂, and I₃ through R₃.

By using Kirchhoff’s Current Law:

∴ I = I₁ + I₂ +I₃    ...(i)

Since the voltage across each resistor is the same V:

I₁ = `V/R_1`

I₂ = `V/R_2`

I₃ = `V/R_3`

Substituting this in equation (i), we get,

I = `V/R_1 + V/R_2 + V/R_3`

= `V(1/R_1 + 1/R_2 + 1/R_3)`

Let R_P be the equivalent resistance of the parallel combination.

By Ohm’s law:

I = `V/R_P`

`V/R_P  = V(1/R_1 + 1/R_2 + 1/R_3)`

⇒ `1/R_P  = 1/R_1 + 1/R_2 + 1/R_3`