Resistors in Parallel
Simple Explanation
When resistors are connected in parallel (across the same two points, providing multiple separate paths for current), the total (equivalent) resistance is found using the reciprocal formula β and the total resistance is always LESS than the smallest individual resistor in the combination.
Why Do We Need It?
Parallel combinations are the second fundamental way resistors combine, and are essential for understanding household electrical wiring, where appliances are connected in parallel so each operates independently.
Formula
Resistors in Parallel
1/Rβ = 1/Rβ + 1/Rβ + 1/Rβ + ...
When resistors are connected in parallel (providing multiple separate paths for current), the reciprocal of the total resistance equals the sum of the reciprocals of each individual resistance.
- Rβ
- β Total (equivalent) resistance of the parallel combination, in ohms
- Rβ, Rβ, Rβ...
- β Resistance of each individual resistor, in ohms
When to use it: Use to find the total resistance of resistors connected across the same two points, providing multiple parallel current paths.
Worked Example
Finding total parallel resistance
Two resistors of 6 Ξ© and 3 Ξ© are connected in parallel. Find the total resistance.
Why Does This Work?
In a parallel circuit, current has multiple independent paths to choose from β adding more parallel paths always makes it EASIER (not harder) for current to flow overall, which is exactly why total resistance decreases (rather than increases) as more resistors are added in parallel, opposite to the series case.
Real-Life Example
Household electrical wiring
Appliances plugged into different outlets in a house all work independently β switching one off does not affect the others.
Household circuits wire appliances in parallel specifically so each receives the full mains voltage and operates independently β this also means adding more appliances (more parallel paths) actually decreases the overall circuit resistance and increases total current drawn from the supply.
Practice
Two resistors of 12 Ξ© and 4 Ξ© are connected in parallel. Find the total resistance.
HardCommon mistake
Forgetting to take the final reciprocal β a very common error is calculating 1/Rβ correctly but then stating that value AS Rβ, instead of taking its reciprocal to find the actual total resistance.
Quick Review
- 1/Rβ = 1/Rβ + 1/Rβ + ... for resistors in parallel.
- Parallel total resistance is always LESS than the smallest individual resistor.
- Parallel components operate independently β one branch failing doesn't stop current in the others.