Learning outcomes
- Apply current and voltage rules in parallel circuits.
- Use the junction rule.
- Calculate equivalent resistance for parallel resistors.
- Explain why adding branches lowers total resistance.
- Compare household parallel wiring with series wiring.
12.1 Branches and common voltage
Components are parallel when connected across the same two nodes. Every parallel branch has the same potential difference because each branch begins and ends at the same electrical potentials.
A 12 V supply therefore places 12 V across each ideal branch. Branch currents may differ because branch resistances differ.
12.2 Junction rule
Charge is conserved at a junction. The total current entering equals the total current leaving. If 5.0 A enters and branch currents are 1.2 A and 2.0 A, the remaining branch carries 1.8 A.
The rule applies at every junction and at every instant in a steady circuit. It is not an energy equation; it expresses conservation of charge.
12.3 Equivalent resistance
Parallel branches provide additional paths for charge, so equivalent resistance is less than the smallest individual branch resistance. For resistors, 1/Rtotal = 1/R1 + 1/R2 + … .
For two resistors, Rtotal = R1R2/(R1 + R2). For n identical resistors of resistance R in parallel, Rtotal = R/n. These shortcuts should be used only for the stated arrangements.
12.4 Why adding a branch increases source current
Adding a parallel branch lowers total resistance. If supply voltage remains fixed, Itotal = V/Rtotal increases. The original branch voltage and current remain approximately unchanged for an ideal source, while the source provides extra current to the new branch.
This differs from adding a series component, which raises resistance and reduces the one common current.
12.5 Household circuits
Household appliances are connected in parallel so each receives the full mains voltage and can be switched independently. If one appliance fails open circuit, others continue operating.
Total current in the supply is the sum of appliance currents. Operating many high-power appliances can exceed cable or circuit ratings, so protective devices are required.
12.6 Identical lamps
Identical lamps in parallel across an ideal source each receive the same voltage as a single lamp and have similar brightness. The total source current is greater because branch currents add.
Battery life may be shorter because more energy is delivered per unit time. Real battery voltage can fall under heavy load due to internal resistance.
12.7 Mixed circuits
For a mixed network, simplify clearly defined series or parallel groups step by step. After finding total current, work backwards to calculate branch voltages and currents.
Do not add resistors directly unless the same current must pass through them. Do not use the parallel formula unless both ends connect to the same two nodes.
Worked examples
Two parallel resistors
6 Ω and 3 Ω in parallel give 1/R = 1/6 + 1/3 = 1/2, so R = 2 Ω.
Branch current
A 12 V supply is across a 4 Ω branch. I = 12/4 = 3 A, regardless of other ideal parallel branches.
Junction
Total current is 7.5 A; two branches carry 2.0 A and 3.1 A. Third current = 7.5 − 5.1 = 2.4 A.
Practical focus
Investigation
Connect two lamps or resistors in parallel. Measure voltage across each branch, branch currents and total current. Verify equal branch voltage and that branch currents sum to total current. Add a third branch and note the increase in source current.
Examination guidance
- Parallel branches have the same p.d.
- Currents add at junctions.
- Equivalent resistance is below the smallest branch resistance.
- Check nodes before declaring components parallel.
Check your understanding
- Find the equivalent resistance of two 8 Ω resistors in parallel.
- A total current of 5.0 A splits into 1.5 A and another branch. Find the second current.
- Why are household appliances connected in parallel?
Answers
- 4 Ω.
- 3.5 A.
- Each receives full mains voltage and can operate independently.
