Resistance in parallel

Resistance in parallel formula

When resistors are connected in parallel, the total or equivalent resistance (R_{eq}) can be calculated using the formula:

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots

We can re-arrange this to find R_{eq}:

R_{eq} = \frac{1}{\left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots\right)}

… where R_1, R_2, R_3, … are the resistances of the individual resistors connected in parallel.

Calculate the resistance of two resistors in parallel

           +-------+           
        +--| 12ohm |--+
        |  +-------+  |
        |             |
--------+             +--------
        |             |
        |  +-------+  |        
        +--| 18ohm |--+
           +-------+

Total resistance, R_{eq}:
- \frac{1}{R_{eq}} = \frac{1}{12\Omega} + \frac{1}{18\Omega}
- \frac{1}{R_{eq}} = \frac{3}{36\Omega} + \frac{2}{36\Omega}
- \frac{1}{R_{eq}} = \frac{5}{36\Omega}
- R_{eq} = \frac{36\Omega}{5} = 7.2\Omega

Increasing the number of resistors in parallel

If we add more resistors in parallel, the equivalent resistance decreases. This is because adding more paths for the current to flow reduces the overall resistance of the circuit (because there are more pathways for the electrons to ‘choose’).

flashcards

Question	Answer
What is the formula for calculating the total resistance (R_{eq}) of resistors in parallel?	\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots
How can you rearrange the parallel resistance formula to solve for R_{eq}?	R_{eq} = \frac{1}{\left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots\right)}
What is the equivalent resistance of a 12\Omega and 18\Omega resistor connected in parallel?	\frac{1}{R_{eq}} = \frac{1}{12\Omega} + \frac{1}{18\Omega} = \frac{5}{36\Omega}, so R_{eq} = \frac{36\Omega}{5} = 7.2\Omega
What happens to the equivalent resistance when you add more resistors in parallel?	The equivalent resistance decreases.
Why does adding more resistors in parallel reduce the overall resistance?	Because adding more paths for current to flow provides more pathways for electrons to ‘choose’, reducing overall resistance.

Resistance in parallel