Kirchhoff's Rules for Multi-Loop Circuits
The junction rule is charge conservation and the loop rule is energy conservation, and together they give enough equations to solve any network. Sign conventions are the only real difficulty.
What a learner can do afterwards
- Assigns currents and signs consistently, then writes independent junction and loop equations
- Solves a two-loop circuit with two sources for every branch current
- Interprets a negative current as a direction guessed the wrong way round
1 · Read
The junction rule is charge conservation: currents entering a junction equal currents leaving it, so the sum in equals the sum out. The loop rule is energy conservation: the voltage changes around any closed loop sum to zero. Crossing a battery from minus to plus counts positive, and crossing a resistor along the current drops the potential by I times R.
Take a single 12.00 V loop with 1.00, 2.00 and 3.00 ohm resistors. The loop rule gives 12 minus I times 1 minus I times 2 minus I times 3 equals zero. That solves to I equals 12 divided by 6, which is 2.00 A through every part.
For a two-loop circuit with two sources, first draw an arrow for each branch current, such as I1, I2 and I3. One junction gives I1 equals I2 plus I3, and the second junction only repeats it, so use one junction equation. Then write one loop equation per loop and solve the three equations together. A negative answer is not a failure: it means the true direction runs opposite your arrow.
Fix a travel direction before writing each loop equation. A resistor crossed with the current contributes minus I R, and crossed against the current contributes plus I R. Multi-loop circuits cannot be reduced to simple series and parallel groups, so write the full set of equations.
One junction equation plus one loop equation per loop gives every branch current, and a minus sign only flips the arrow.
More to read
10.1 Electromotive Force
A battery's electromotive force is the work per charge it supplies, the push that keeps current looping around a circuit. Terminal voltage matches the emf only when no current flows; under load, internal resistance steals a share. Think of emf as the lift in a water loop and internal resistance as a narrow stretch of pipe inside the pump itself. The lift sets the circulation, but the pipe always takes its cut.
10.3 Kirchhoff's Rules
Kirchhoff's junction rule says currents entering a node equal those leaving, which is charge conservation in disguise. The loop rule says the voltage rises and drops around any closed path sum to zero, which is energy conservation. Solving a two loop circuit means labelling every branch current, writing one junction equation and two loop equations with steady sign rules, then solving the set. A negative answer is not an error: it means that branch flows opposite to the guessed arrow.
10.2 Resistors in Series and Parallel
Resistors in series share one current and split the voltage, so their resistances add directly. Resistors in parallel share one voltage and split the current, so their reciprocals add. Shrinking a network step by step with these rules is always worth trying before the full Kirchhoff treatment. Fewer branches mean fewer equations and fewer sign chances to slip.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.