# Parallel Resistors

fact

Two resistors are in parallel if they are directly connected to each other at both ends. If other things are also connected at their end points it doesn't matter.

fact

Two resistors in parallel have the same voltage drop across them.

fact

You can combine two parallel resistors into one equivalent resistor just like with series resistors.The equivalent resistor's resistance is given by one of the following formulas (they're all the same, just written in different ways):
\(R_{eq} = \frac{R_1 R_2}{R_1 + R_2}\)
\(R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}\)

example

Combine the following resistors into a single, equivalent resistor

We just use our handy dandy formula above to find: \(R_{eq} = \frac{R_1R_2}{R_1 + R_2} = \frac{100 \cdot 50}{100+50} \approx 33.3\Omega\)fact

When two resistors are in parallel their equivalent resistance is always less then either of their resistances.

fact

If two parallel resistors have the same resistance then their equivalent resistance is half their original value.

fact

If three or more resistors are in parallel you can combine them into a single equivalent resistor by combining any two into an equivalent resistor and then continuing until you have just one equivalent resistor left (see an example below).

example

Combine all 4 resistors below into a single equivalent resistor.

First let's combine resistors R1 and R2 to get \(R_{12} = \frac{100\times 500}{100 + 500} = 83.33\Omega\) Now we combine R3 and R4 to get \(R_{34} = \frac{200 \times 500}{200 + 500} = 142.86\Omega\) And finally we can combine \(R_{12}\) and \(R_{34}\) to get \(R_{eq} = \frac{83.33 \times 142.86}{83.33 + 142.86} = 52.63\Omega\)
practice problems