Resistors in Parallel

It probably won't surprise you that we now want to calculate the equivalent resistance of two resistors in parallel. From a comparison with the results for capacitance circuits you can probably also guess what the result will be.

Two resistors are in parallel, if they are connected across the same potential drop. This means that, generally, different currents will flow through them. One example would be two lamps or two electrical appliances on the same household circuit.

Here the potential across each resistor is the same V₁ = V₂ = V
And now the two current flowing through the two resistors have to add up to the total current I₁ + I₂ = I
If a single resistor R_p has to be equivalent to R₁ and R₂ in parallel, it must have the same current I for the same potential drop V across its ends, \[ \rm \mathbf{\frac{V}{R_{p}} = I} \]

Combining these three equations we arrive at \[ \rm \mathbf{\frac{V}{R_{p}} = I = I_{1} + I_{2} = \frac{V_{1}}{R_{1}} + \frac{V_{2}}{R_{2}} = \frac{V} {R_{1}} + \frac{V}{R_{2}}} \]

canceling out the factor V:

1	=	1	+	1

R_p		R₁		R₂

Another way to write this is,

R_p =	R₁ $\cdot$ R₂

	R₁ + R₂

For n resistors this generalizes to:

1	=	1	=	1	+ ... +	1

R_p		R_i		R₁		R_n

Adding another resistor in parallel always reduces the total resistance.