{"id":572,"date":"2026-07-15T03:28:09","date_gmt":"2026-07-15T03:28:09","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=572"},"modified":"2026-07-15T03:28:11","modified_gmt":"2026-07-15T03:28:11","slug":"kirchhoffs-law","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/electromagnetism\/kirchhoffs-law\/","title":{"rendered":"Kirchhoff’s Laws (Current & Voltage)"},"content":{"rendered":"\n
Definition<\/div>

\nKirchhoff’s law comes in two parts: the current law says the currents arriving at any junction equal those leaving it, and the voltage law says the potential differences around any closed loop sum to zero. Written compactly, \u03a3I = 0 and \u03a3V = 0 \u2014 conservation of charge and conservation of energy, applied to circuits.\n<\/p><\/div>\n\n

Somewhere in every physics course there is a circuit that breaks you. It has two batteries pointing at each other, a resistor shared between two loops, and no amount of squinting turns it into “series” or “parallel”. Ohm’s law, which has served faithfully until this exact moment, simply has nothing to say about it.<\/p>\n\n

That circuit is not a trick. It is the ordinary case \u2014 your phone charger, a car’s electrical system, the power supply on your desk \u2014 and it is what Kirchhoff’s laws were built for. Two rules. Both of them things you already believe.<\/p>\n\n

What Is Kirchhoff’s Law?<\/h2>\n\n

Kirchhoff’s law is a pair of rules for electrical circuits: the current law (KCL) states that the sum of currents at any junction is zero, and the voltage law (KVL) states that the sum of potential differences around any closed loop is zero.<\/strong> Together they let you solve any circuit made of ordinary components, however tangled.<\/p>\n\n

Here is the thing worth noticing. Neither law is really about electricity.<\/p>\n\n

The current law is a statement that charge does not pile up or vanish at a wire junction \u2014 whatever arrives must leave, because there is nowhere else for it to go. The voltage law is a statement that if you walk around a closed loop and return to where you started, you are back at the same potential. You cannot gain energy by walking in a circle. Both are conservation laws wearing a circuit diagram as a disguise.<\/p>\n\n

That is why they are so powerful. Conservation of charge and conservation of energy do not care whether your resistors are in series, in parallel, or in a knot no textbook has a name for.<\/p>\n\n

A Note on the Names<\/h3>\n\n

The same two laws travel under several names, which trips people up when they compare sources. The current law is also the first law<\/strong>, the junction rule<\/strong>, the node rule<\/strong>, or KCL<\/strong>. The voltage law is also the second law<\/strong>, the loop rule<\/strong>, the mesh rule<\/strong>, or KVL<\/strong>. Same physics, different textbook.<\/p>\n\n

Gustav Robert Kirchhoff announced both laws in 1845<\/strong>, aged 21 and still a student at Albertus University of K\u00f6nigsberg. He developed them in the mathematics-physics seminar run by Franz Neumann and Carl Jacobi, which he attended from 1843 to 1846, and they extended Georg Ohm’s work from single components to whole networks. He graduated in 1847, and later did major work in spectroscopy and black-body radiation. The MacTutor archive at St Andrews<\/a> carries the fuller biography.<\/p>\n\n

\n \"Gustav\n
Gustav Robert Kirchhoff (1824-1887). He worked out both circuit laws as a student, before he had a doctorate.<\/figcaption>\n<\/figure>\n\n

The Kirchhoff’s Law Formulas<\/h2>\n\n

Both laws are sums set equal to zero. That is the whole of the notation.<\/p>\n\n

Kirchhoff’s Current Law (KCL) \u2014 the junction rule:<\/strong><\/p>\n\n

\u03a3I = 0 (at any junction)<\/div>\n\n

The algebraic sum of all currents at a junction is zero. Count currents flowing in as positive and currents flowing out as negative. Equivalently, and more usefully in practice:<\/p>\n\n

\u03a3I(in) = \u03a3I(out)<\/div>\n\n

Kirchhoff’s Voltage Law (KVL) \u2014 the loop rule:<\/strong><\/p>\n\n

\u03a3V = 0 (around any closed loop)<\/div>\n\n

The algebraic sum of all potential differences around any closed loop is zero. Sources count as rises, resistors count as drops, and the signs are set by the direction you walk.<\/p>\n\n

Every Symbol, With Its SI Unit<\/h3>\n\n
\n\n\n\n\n\n\n\n\n\n\n
Symbol<\/th>\nQuantity<\/th>\nSI unit<\/th>\n<\/tr>\n<\/thead>\n
I<\/strong><\/td>\nElectric current \u2014 rate of charge flow<\/td>\nampere (A)<\/td>\n<\/tr>\n
V<\/strong><\/td>\nPotential difference across a component<\/td>\nvolt (V)<\/td>\n<\/tr>\n
\u03b5<\/strong><\/td>\nEMF \u2014 the voltage a source supplies<\/td>\nvolt (V)<\/td>\n<\/tr>\n
R<\/strong><\/td>\nResistance<\/td>\nohm (\u03a9)<\/td>\n<\/tr>\n
Q<\/strong><\/td>\nElectric charge<\/td>\ncoulomb (C)<\/td>\n<\/tr>\n
\u03a3<\/strong><\/td>\n“The sum of” \u2014 add every term, with its sign<\/td>\n\u2014<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n

Notice that Kirchhoff’s laws never mention resistance. They constrain currents and voltages only. Every loop equation you write becomes solvable because you substitute V = IR<\/strong> for each resistor \u2014 so Ohm’s law is the engine and Kirchhoff’s laws are the steering. If you want to check a single V, I or R value while you work, our Ohm’s Law Calculator<\/a> handles that one substitution instantly; the loop bookkeeping is still yours.<\/p>\n\n

Kirchhoff’s Current Law: The Junction Rule<\/h2>\n\n

Kirchhoff’s current law states that the total current flowing into any junction equals the total current flowing out of it.<\/strong> Charge is conserved, and a junction is just a point in a wire \u2014 it has no storage.<\/p>\n\n

Think of a road fork. Eight cars per minute arrive; eight cars per minute must leave, split however the roads dictate. Nothing accumulates at the fork itself, because a fork is not a car park.<\/p>\n\n<\/rect>KIRCHHOFF’S CURRENT LAW AT A JUNCTION<\/text><\/line><\/line><\/line><\/line><\/polygon><\/polygon><\/polygon><\/polygon><\/circle><\/circle>I1<\/tspan> = 5 A<\/tspan><\/text>in<\/text>I2<\/tspan> = 3 A<\/tspan><\/text>in<\/text>I3<\/tspan> = 2 A<\/tspan><\/text>out<\/text>I4<\/tspan> = 6 A<\/tspan><\/text>out<\/text>5 A + 3 A = 2 A + 6 A so \u03a3I = 0<\/text>charge in<\/text>charge out<\/text><\/svg>\n\n

Kirchhoff’s current law at a junction: 5 A and 3 A arrive, 2 A and 6 A leave. Eight amperes in, eight amperes out.<\/p>\n\n

Write it with signs and the two forms agree. Taking “in” as positive: (+5) + (+3) + (\u22122) + (\u22126) = 0. Taking in equals out: 5 + 3 = 2 + 6. Use whichever you find less error-prone \u2014 most people find “in equals out” harder to get wrong.<\/p>\n\n

Why It Has to Be True<\/h3>\n\n

Charge is conserved, and a junction has no capacity to store it. If more charge arrived than left, charge would accumulate at that point in the wire \u2014 building an electric field that would immediately push back and stop the imbalance. In practice this happens so fast, and at such vanishingly small charge, that the junction is always balanced for any circuit you will meet in a lab.<\/p>\n\n

That “in practice” hides a real limit, and we return to it below. A capacitor plate is<\/em> a node where charge genuinely accumulates.<\/p>\n\n

Kirchhoff’s Voltage Law: The Loop Rule<\/h2>\n\n

Kirchhoff’s voltage law states that the sum of all potential differences around any closed loop is zero.<\/strong> Every rise is paid for by a drop somewhere else in the loop.<\/p>\n\n

Picture a walk through a hilly town that ends at your front door. Climb, descend, climb again, wander as much as you like \u2014 the moment you are back at the door, your net change in altitude is exactly zero. It has to be. It is the same door.<\/p>\n\n

Electric potential behaves identically. A battery lifts charge to a higher potential; resistors let it fall back down. Return to your starting point and the rises and drops must cancel exactly.<\/p>\n\n

The Four Sign Cases \u2014 This Is Where Marks Are Lost<\/h3>\n\n

KVL is arithmetic. The only difficulty is the sign of each term, and there are exactly four cases to know.<\/p>\n\n<\/rect>THE FOUR KVL SIGN CASES \u2014 THE ONLY THING YOU MUST GET RIGHT<\/text><\/rect>RESISTOR \u2014 WALK WITH THE CURRENT<\/text><\/line><\/polygon>I<\/text><\/line><\/rect><\/line>R<\/text><\/line><\/polygon>walk<\/text>-IR<\/text>potential DROPS<\/text><\/rect>RESISTOR \u2014 WALK AGAINST THE CURRENT<\/text><\/line><\/polygon>I<\/text><\/line><\/rect><\/line>R<\/text><\/line><\/polygon>walk<\/text>+IR<\/text>potential RISES<\/text><\/rect>BATTERY \u2014 WALK FROM MINUS TO PLUS<\/text><\/line><\/line><\/line><\/line>–<\/text>+<\/text><\/line><\/polygon>walk<\/text>+\u03b5<\/text>potential RISES<\/text><\/rect>BATTERY \u2014 WALK FROM PLUS TO MINUS<\/text><\/line><\/line><\/line><\/line>+<\/text>–<\/text><\/line><\/polygon>walk<\/text>-\u03b5<\/text>potential DROPS<\/text><\/svg>\n\n

The four sign cases for Kirchhoff’s voltage law. The sign depends on the direction you walk, not on the component.<\/p>\n\n

The rule in one line: the sign is decided by your direction of travel, not by the component.<\/strong> Walk downhill and the term is negative; walk uphill and it is positive.<\/p>\n\n