ΣI = 0 at a junction · ΣV = 0 round a loop · signs · two-loop circuits
Ohm's law handles one resistor at a time. The moment a circuit branches, you need two rules instead: charge cannot pile up where wires meet, and voltage cannot gain or lose anything by going round in a circle. Those two sentences solve every circuit in school physics.
This page covers the junction rule, the loop rule, the sign convention that makes them usable, and how to solve a circuit that series-and-parallel thinking cannot touch. By the end you'll be able to:
Put two batteries in one circuit and every series-parallel trick fails at once. Kirchhoff is what is left, and boards know it:
Searched as: kirchhoff's law, junction rule, loop rule, KCL KVL, two loop circuit, sign convention, wheatstone bridge.
Predict the current in the middle branch, lock it, then change the batteries and the resistor. The bench solves both loop equations on every frame. 🟢 real two-loop solve
Guess, then lock — the bench keeps its answer until you commit.
Set the two cells and the middle resistor — then check both rules against the numbers.
The Circuit Bench above is finished and real — it solves the two loop equations on every frame rather than showing a stored answer. The teaching sections, the glossary, the worked examples and the question bank are being written now.
Before any algebra, name the rule the picture is asking for. Six situations — your answers build the table underneath.
Three wires meet at a point. Which rule answers it?
Read the picture, then name the rule.
Two rules and one law cover every circuit in the syllabus. The skill is choosing, not calculating.
Two currents arrive at the node. Dial the third until the junction balances — 3 attempts.
Set the third current so nothing collects at the node.
Each one breaks a rule somewhere. Tap the broken idea.
One amp a second would have to collect at the node forever. Charge does not pile up there, so the outgoing currents must add to three.
You have returned to the same point, so you are at the same potential. A closed walk that gains five volts would be free energy.
A resistor takes energy from the charge, not the charge itself. With nowhere to branch, the same current runs all along the wire.
Thinking beats memorising. Sort every circuit claim into three boxes.
You are walking round a loop in the marked direction. Each step, say what the term is. The signs are the whole difficulty of this chapter.
Crossing a resistor along the current is a drop. Entering a cell at the short plate is a rise.
One cell, two resistors in series. Pick the voltage you need across the lower one, then find the pair that gives it.
The loop rule gives V₂ = V × R₂/(R₁ + R₂), with the supply fixed at 12 V.
Read the scene. Which rule is doing the explaining?
Sockets sit in parallel across the supply, so each gets the full voltage and draws what it needs. The junction rule adds those draws up in the main cable — which is what the main fuse is sized for.
A series string is one loop. Break it anywhere and there is no closed path, so the current everywhere is zero — which is why modern strings are wired in parallel groups.
Two cells in one loop is the classic Kirchhoff case. The loop rule gives (ε₁ − ε₂) = I(r₁ + r₂ + R), and the small internal resistances are why the current is enormous.
In plain terms: one rule says charge is conserved where wires meet, the other says energy is conserved round any closed path. Everything else is bookkeeping.
Charge cannot collect at a point in a wire, so whatever arrives each second must leave each second: ΣI = 0. Count currents into a junction as positive and out as negative, or the other way — it only has to be the same for the whole equation. For a circuit with n junctions, exactly n − 1 of these equations are independent.
Walk any closed path and add every rise and every drop. You end where you began, at the same potential, so the total is zero. A cell contributes +ε when you walk into it from its short plate, and a resistor contributes −IR when you cross it along the current. Both signs flip if you walk the other way.
You do not need to know which way the current really runs. Draw an arrow, any arrow, write the equations with it, and solve. If the answer comes out negative, the size is still right and the arrow was simply backwards. This is the part students distrust and it is the part that makes the method work on circuits nobody can read by eye.
A branch with an open switch carries zero, but the loop rule still holds for the loops that remain. A cell being charged appears with its EMF opposing the current, which is a minus where you expected a plus. And a balanced Wheatstone bridge is the case where the middle branch carries exactly nothing — not because it is broken, but because both ends sit at the same potential.
Exams reward the method. Work it one step at a time — read the thought, predict the line, then reveal it.
Three small labs, each one native to these rules.
Six wrong ideas that survive because they sound sensible. Tap a card to bust the myth.
Still a student at Königsberg, Gustav Kirchhoff published the two rules while working through Ohm's ideas for networks nobody could simplify. They were a seminar exercise.
He set the rules out as a general method for any network of branches, which is the form every textbook still teaches. The telegraph industry had its mathematics.
With Bunsen he founded spectroscopy and later explained black-body radiation. The circuit rules were the work of his early twenties.
SPICE and its descendants build one junction equation per node and one loop equation per mesh, then solve the matrix. Chip design runs on this page.
Load-flow programs solve the same two statements across thousands of substations every few seconds, to decide what each power station must produce next.
Sockets in parallel, each drawing what it needs; the junction rule adds those draws in the main cable, and that sum is what the fuse is chosen for.
Two cells in one loop, differing by a few volts across a fraction of an ohm. The loop rule is the reason the current is measured in hundreds of amps.
Inside it, dividers set reference voltages and feedback loops hold them steady — all of it designed with these two rules and nothing more exotic.
One sentence each: what the junction says, and what the loop says.
Drag the slider. Same two rules, six circuits, from one loop to thousands of nodes.
The circuit grows; the two sentences do not change at all.
Each one is a build and a measurement — the measuring is what turns a demo into a project.
Build: one cell, three bulbs, two of them side by side so the current splits.
Measure: the current in the main wire and in each branch. The two branch readings should add to the main one.
Build: a cell and three resistors in one series loop.
Measure: the voltage across every component in turn. The drops should add to the cell's reading.
Build: two cells of different voltage, each with its own resistor, sharing a third.
Measure: all three branch currents. Predict them first with the two rules and compare.
Build: a cell, a variable resistor and a meter.
Measure: terminal voltage against current for several settings; the slope of the line is −r.
Build: four resistors in a diamond with a meter across the middle.
Measure: adjust one until the middle reads zero, then check R₁/R₂ = R₃/R₄.
Build: five bulbs in series, then the same five in parallel.
Measure: brightness and total current for each, and unscrew one bulb in both.
Straight answers, each one complete on its own:
The junction rule says the currents arriving at any junction add up to the currents leaving it, because charge cannot collect there. The loop rule says that if you walk round any closed path and add every rise and drop, the total is zero, because you finish at the same potential you started from.
The moment the circuit will not simplify. Two cells in different branches, or a bridge with a component across the middle, cannot be reduced by series and parallel rules at all. Kirchhoff's method works on any network, which is why it is the general answer.
No, and that is the point of the method. Draw them any way you like, keep them consistent through every equation, and solve. A current that comes out negative is simply running opposite to the arrow you drew; its size is correct and no rework is needed.
Pick a direction to walk and keep it. Crossing a resistor along the current direction is a drop, so the term is −IR; against it, +IR. Entering a cell at its short plate and leaving at the long one is a rise, +ε; the other way is −ε. Any consistent set works.
One per unknown current. For n junctions, n − 1 of the junction equations are independent; the rest come from loops, choosing loops that each include at least one branch the others do not. Three branch currents need three equations, typically one junction and two loops.
No. It is a rise when you walk through it from its negative plate to its positive one, whichever way the current happens to run. A cell being charged by a stronger one in the same loop will appear with its EMF opposing the current, and its term carries the sign that fact gives it.
A voltmeter measures a loop term, the difference between two points, so it connects across the component and must draw almost no current. An ammeter measures a junction term, the current through a branch, so it has to sit in that branch and must have almost no resistance.
The two ends of the middle branch reach the same potential, so the loop containing it has no driving term and it carries no current. The balance condition R₁/R₂ = R₃/R₄ follows from the two rules alone, and because it is a ratio, the supply voltage cancels out entirely.
Spot these before the exam does.
Seven question formats, the way LLOS.ai serves them — multiple choice, multiple-correct, fill-in-the-blank, match, sequence, read-think-connect, and write-your-own. Every question has layered hints: a quick nudge, the reasoning, then a deeper connection — so a wrong answer opens a door, never a dead end. 🟢 received from a board-tagged question bank · seed toward 2,000
Pick your board — the set re-tunes to its wording and emphasis. Competitive draws the JEE / NEET / Olympiad lane.
Four circuits, no numbers yet. Tap them in order, biggest middle current first. Then read what the two rules give — and see what fooled you.
A 24 V cell on the left, 6 V on the right, ordinary 10 Ω resistors everywhere.
Two 12 V cells, one each side, and the same 10 Ω resistors everywhere.
Two 12 V cells again, but the shared middle branch is only 2 Ω.
Two 12 V cells, and a 200 Ω resistor in the middle branch.
Tap the four in the order you believe, biggest middle current first.
Nothing is lost where wires meet, and nothing is gained by walking in a circle. Every circuit you will ever solve is those two sentences, written down carefully.