Short on time? This page is the recap. For the derivation, the figures and the worked examples, use the full study guide.
Recap
- A junction is a point where three or more wires meet. A branch runs between two junctions and carries one current throughout.
- Charge cannot build up at a junction in a wire, so the charge arriving per second equals the charge leaving per second.
- In calculus form: ∮J·dA = −dq_inside/dt, and dq_inside/dt = 0 for a junction.
- The rule also applies to any closed region of a circuit. Use this as a check.
- It holds at every instant, even while currents change (for example while a capacitor charges), because charge collects on capacitor plates, not at junctions.
Key relationships
| Idea | Statement |
|---|---|
| Junction rule | ΣI_in = ΣI_out, or ΣI = 0 with “in” positive |
| Current and charge | I = dq/dt, so equal currents mean equal charge per unit time |
| Independent equations | N junctions give N − 1 independent junction equations |
| Unknowns | One current per branch; B branches need B equations in total |
| Node form | Current in a resistor = (V_start − V_end)/R, then apply ΣI_in = ΣI_out at each unknown node |
Method for a multi-loop circuit
- Label junctions and one current per branch, each with an assumed direction.
- Write N − 1 junction equations.
- Write loop equations for the remaining unknowns (Topic 11.6).
- Solve, then interpret any negative signs.
- Check with a closed region, an equivalent resistance or an energy balance.
Assumptions
- Batteries, wires and meters are ideal unless stated.
- Junctions store no charge.
- Circuits with batteries of different potential differences joined in parallel are outside the unit’s boundary.
Mistakes to avoid
- “Current is used up.” The same current enters and leaves each element.
- Equal splitting. Current divides equally only between identical branches.
- Redrawing arrows mid-solution. Keep your assumed directions; fix the sign at the end.
- N junction equations. Only N − 1 are independent.
- One current per resistor. Use one per branch.
Quick self-check
- Currents of 0.8 A enter a junction along one wire, and 0.3 A and 0.2 A leave along two others. What is the current in the fourth wire? (0.3 A, leaving)
- In 0.50 s, 1.5 C of charge passes along a wire into a junction. What current does this wire carry? (3.0 A)
- You assume I₄ flows into a junction and get I₄ = −0.2 A. What does this mean? (0.2 A flows out of the junction)
Next: practice questions.