Kirchhoff's Laws
Conservation of charge and energy, applied to circuits.
Lesson at a glanceShow the key points
- Junction law: current in = current out.
- Loop law: the potential changes round any closed loop sum to zero.
- Signs: resistor along current: drop iR; cell − to +: rise ℰ.
- A negative answer means the current flows opposite to the guess.
Two simple rules, one about charge and one about energy, are enough to solve any circuit of cells and resistors.
The junction law
The total current flowing into any point of a circuit equals the total current flowing out. Equivalently, counting currents toward the point as positive and away as negative, their algebraic sum is zero.
Why: no point in a circuit keeps piling up charge or keeps producing it, so whatever arrives must leave. It is conservation of charge.
The loop law
Going all the way round any closed loop, the potential differences add up to zero. Walk round the loop in one direction and list each change: count a drop in potential as positive and a rise as negative (or the other way round, consistently).
- Through a resistor in the direction of its current: drop iR. Against the current: rise iR.
- Through a cell from − to +: rise ℰ. From + to −: drop ℰ.
Why: the electrostatic force is conservative, so its work round a closed path, and hence the net change of potential, is zero. You end where you started.
Cell ℰ₁ = 10 V in series with R₁ = 2 Ω and cell ℰ₂ = 4 V in series with R₂ = 4 Ω both feed the top of a shared resistor R₃ = 4 Ω, whose bottom joins both cells’ negative terminals. Find all three currents. (This is the widget’s starting circuit.)
- 1Junction law: current in = current out (charge conservation).
- 2Loop law: potential changes round a closed loop sum to zero (conservative field).
- 3Resistor along the current: drop iR; cell − to +: rise ℰ.
- 4Negative results mean the real current is opposite to the guess.
Check yourself
Keep exploring
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