### Derivation of the Parallel Resistance Formula

It’s easy to see where the series resistance formula comes from; add a resistor, add its resistance. It’s more difficult to see where the parallel resistance formula comes from. However, the formula can be derived using a simple combination of KCL and Ohm’s law. Take the simple circuit shown in Figure 11. KCL says that the total current in the circuit must be equal to the currents I_{1} and I_{2}.

*From* KCL:

Using Ohm’s law gives equations for I_{s}, I_{1}, and I_{2}.

*From Ohm’s Law:*

Plug these three equations back into the first equation and solve for R_{1}||R_{2} to derive the formula for two parallel resistors.

Which gives us…

There you have it, a technical explanation of the parallel resistance formula.

Written by Ryan Eatinger (reatinge@ksu.edu). Thank you!