Kirchhoff's Rules

Series and parallel tricks are lovely while they last — and they last about two batteries. Add a second source somewhere unexpected, or a resistor bridging two branches, and suddenly no pair of elements in the circuit is "in series" or "in parallel" with anything: the reduction shortcuts simply have nothing to grab. Yet the electrons are not confused. Currents settle instantly into one definite pattern, the same every time you connect the terminals. Something is deciding — and it is not algebra tricks. Two physical laws are deciding, and this article is about wielding them. Charge cannot pile up at a junction, so the currents into any node must balance — that is the whole content of the junction rule. And in a steady circuit the electric potential is a single-valued function of position, so walking any closed loop must bring your running total of voltage rises and drops back to exactly zero — that is the loop rule. Kirchhoff wrote both down in 1845, as a 21-year-old student, and they have been the constitution of circuit analysis ever since: every network, however gnarled, surrenders to them by systematic bookkeeping. You are ready for this. The track has already given you the loaded battery ( $V = \varepsilon - Ir$ ) and the resistor ( $V = IR$ ); what remains is the discipline of signs and the payoff of trusting it — including the Wheatstone bridge, a circuit whose entire genius is a reading of zero, and an honest look at when the loop rule is allowed to fail.

Key takeaways

The junction rule $\sum I_{\text{in}} = \sum I_{\text{out}}$ is conservation of charge at a node — steady currents cannot stockpile charge anywhere.
The loop rule $\sum \Delta V = 0$ around any closed path is the single-valuedness of potential — the electrostatic field does no net work around a loop.
Method: name the branch currents, guess directions freely, write junction and loop equations, and solve. A negative answer means the true current runs opposite your arrow — nothing is ruined.
A circuit with $B$ branches and $J$ junctions needs $J-1$ junction equations and $B-J+1$ loop equations: exactly $B$ equations for $B$ unknowns.
The Wheatstone bridge turns the rules into a precision instrument: at balance the galvanometer reads zero and $R_x = R_2R_3/R_1$ , independent of the supply.
The loop rule is conditional: it holds only while no changing magnetic flux threads the loop — the honesty clause that leads to Faraday and RL Circuits.
KEY IDEA: two conservation laws, applied with disciplined signs, solve every circuit that exists.

The core insight

Kirchhoff's rules are charge conservation and energy-per-charge conservation written as circuit bookkeeping: currents balance at every junction, and potential changes sum to zero around every loop — enough, always, to determine every current in the network.

Where this lives today

Every integrated circuit designed this century was signed off by Kirchhoff's rules running at industrial scale. SPICE and its descendants — the simulators inside every chip-design flow — assemble exactly the node equations of this article (as modified nodal analysis) for networks of billions of elements, then solve the resulting sparse linear systems millions of times over as the waveforms evolve. When a phone's power-management chip is verified against brown-out, or a CPU's clock tree is timed, the "verification" is Derivation B's bookkeeping executed by machine at a scale no human could survey, but with not one idea added. The same rules run the continental scale. Power-flow studies that schedule national grids are Kirchhoff's equations on networks whose "wires" are 500 kV corridors hundreds of kilometres long — with the honest complication that at 50–60 Hz and continental spans, the lumped treaty needs care, and line models sprout the inductances and capacitances the loop rule's fine print warned about. And the Wheatstone bridge never retired: strain gauges, platinum thermometers, pressure sensors and load cells overwhelmingly report through bridge circuits, because a century and a half later, detecting a small departure from zero is still the most precise measurement electronics knows how to make.

The physical picture

Picture the circuit as a road network for charge. Junctions are intersections: cars flow through them, but no intersection stores cars — whatever drives in per second drives out per second. That is the junction rule, and it is not an approximation. In a steady state, any node that accumulated charge would build an electric field repelling further arrivals, choking the imbalance off in nanoseconds; the Continuity Equation article expresses the same statement at field level as $\nabla\!\cdot\!\mathbf{J} = 0$ for steady currents. The circuit version is that law integrated over a small bag enclosing the node. The loop rule is an altitude statement. In a steady circuit, every point has one definite potential — a single number, like elevation on a landscape. Walk any closed tour of the landscape and your net change in altitude is zero, no matter the route: climbs through batteries and descents through resistors must cancel exactly. The track proved the underlying fact in Curl of the Electric Field: the electrostatic field has $\oint \mathbf{E}\cdot d\boldsymbol{\ell} = 0$ , so potential is well defined. The loop rule is that theorem wearing work boots.

Figure

The circuit that defeats series-parallel thinking — two sources, three branches, two junctions: nothing is in series or parallel with anything, yet two rules determine every current

Look at the figure's circuit honestly for a moment. Is $R_1$ in series with $R_3$ ? No — junction a sits between them, siphoning current into $R_2$ 's branch. Is $R_1$ parallel to $R_2$ ? No — their left and right ends connect through different batteries. The reduction vocabulary simply does not parse this network. But count what the rules give us: three unknown currents; one independent junction statement ( $I_1 = I_2 + I_3$ at a — the statement at b repeats it); two independent loops. Three equations, three unknowns, one guaranteed answer. That counting — $J-1$ junction equations plus $B-J+1$ loop equations for $B$ branches and $J$ junctions — always comes out exactly right, which is why the method never jams. The second picture to carry is the null instrument. Arrange four resistances in a diamond, drive one diagonal, and hang a galvanometer across the other. In general the meter deflects; but tune one arm and there is a magic ratio at which both meter terminals sit at the same altitude — zero current, zero deflection, and an unknown resistance measured by pure comparison. That is the Wheatstone bridge, and we derive it below as the rules' classic payoff.

Figure

The Wheatstone bridge — a diamond of four resistances with a galvanometer across the middle: at balance the meter reads exactly zero and the unknown follows from a ratio

The two rules, stated for work

The junction rule is bookkeeping at a point. At any node of the circuit,

\sum I_{\text{in}} = \sum I_{\text{out}} \qquad (1)

where: each $I$ — a branch current, counted with the direction of its arrow. Physically this is charge conservation plus the refusal of a steady state to store charge at a node: the Continuity Equation article's $\nabla\!\cdot\!\mathbf{J} = 0$ , integrated over the junction. It fails only where charge genuinely does accumulate by design — a capacitor plate is exactly such a place, which is how RC Circuits will extend, not break, this rule. The loop rule is bookkeeping around a tour. Around any closed loop, traversed in a chosen direction,

\sum_{\text{loop}} \Delta V = 0 \qquad (2)

where: $\Delta V$ — the potential change across each element as you cross it, signs and all: $-IR$ crossing a resistor with the current, $+IR$ against it; $+\varepsilon$ crossing a source from $-$ terminal to $+$ , $-\varepsilon$ the reverse (and $-Ir$ for its internal resistance either way, per EMF and Internal Resistance). Its physical root is that potential is single-valued — the altitude picture — which the track earned by proving $\oint\mathbf{E}\cdot d\boldsymbol{\ell} = 0$ for electrostatic fields. Equation (2) carries fine print that equation (1) does not, and honesty demands it now rather than in a footnote. Single-valued potential requires curl-free fields, and Faraday discovered the exception: a magnetic flux changing through the loop drives a circulating EMF, $\oint\mathbf{E}\cdot d\boldsymbol{\ell} = -d\Phi_B/dt \neq 0$ . The classroom demonstration made famous by Walter Lewin — two voltmeters clipped to the same two points of a loop threaded by a changing flux, honestly reading different values — is not a paradox but this clause in action: with flux changing, "the voltage between a and b" stops being a well-defined single number. Circuit analysis survives by the lumped-element discipline: confine every changing flux inside declared components (inductors), assign each a terminal voltage $L\,dI/dt$ , keep stray loop areas small, and equation (2) rules again. That treaty is signed in Faraday's Law of Induction and first exploited in RL Circuits; at radio frequencies, where every centimetre of wire is an appreciable flux loop, its terms get strict — the deep reason "Kirchhoff lies at high frequency."

Derivation AThe two rules from two conservation laws

We want both rules as consequences of physics the track has already established, not as postulates — junction rule from charge conservation, loop rule from the conservative character of the electrostatic field — and we want the exact boundary where each can fail. The strategy is to take the field-level statements and integrate them over circuit-sized regions: a small closed surface swallowing a node, and a closed path along the wires. Watch for the two escape hatches, because both are physically real and both become circuit elements in later articles.

Step 1. Enclose a junction in an imaginary closed surface $S$ . Charge conservation says the net current flowing out of $S$ equals the rate the enclosed charge decreases: $\oint_S \mathbf{J}\cdot d\mathbf{a} = -\,dQ_{\text{enc}}/dt$ .

Step 2. In steady state, $Q_{\text{enc}}$ is constant — any transient pile-up self-destructs, because accumulating charge generates a field that repels further arrivals, relaxing on the conductor's internal timescale (femtoseconds in copper, per Conductors in Electrostatic Equilibrium). So $\oint_S \mathbf{J}\cdot d\mathbf{a} = 0$ : total current in equals total current out. Cutting the surface's crossings into the individual wires gives equation (1). Why the capacitor is the licensed exception: put one capacitor plate inside $S$ and $dQ_{\text{enc}}/dt$ is the whole point — the branch "current into the plate" then equals the plate's charging rate, which is exactly how the junction rule generalises in RC Circuits rather than failing.

Step 3. For the loop rule, recall the electrostatic circulation theorem from Curl of the Electric Field: $\oint \mathbf{E}\cdot d\boldsymbol{\ell} = 0$ for fields sourced by charges. This is precisely the statement that a potential function exists.

Step 4. Walk a closed circuit loop and split the tour into element crossings. Across each resistor, the field does work $-IR$ per unit charge in the direction of current; across each source, chemistry (or another non-electrostatic agent, the $\mathbf{f}_s$ of Electromotive Force) lifts the charge by $\varepsilon$ while internal resistance taxes $Ir$ . Summing all potential changes around the tour and using single-valuedness — you return to the altitude you started from — gives equation (2).

Step 5. The escape hatch here is Faraday's: if a changing magnetic flux $\Phi_B$ threads the loop, the circulation is $-d\Phi_B/dt$ instead of zero, and the "sum of drops" acquires a source term belonging to no single element. The lumped-circuit treaty quarantines all such flux inside inductors so that (2) survives as bookkeeping. Steady circuits — this article's domain — satisfy it exactly. Verified: both integral statements reduce to the quoted circuit forms under the stated conditions; consistency of the sign conventions is checked numerically in Derivation B, where the resulting solution balances the power ledger to the last digit.

The method — and one circuit, beaten thoroughly

Systematic beats clever. The procedure that never jams: label every branch current with a name and an arrow you choose freely; write junction equations at all nodes but one; pick independent loops (each must visit at least one branch no previous loop used) and walk each with disciplined signs; solve the linear system; and read negative answers calmly — a minus sign means the real current opposes your arrow, full stop. The circuit in the hero figure is the perfect proving ground because series-parallel vocabulary cannot even begin it.

Go deeper · Derivation B · 2 minA two-source circuit solved end to end

Derivation BA two-source circuit solved end to end

We want all three currents in the hero circuit: a 12 V source with $R_1 = 4\ \Omega$ in the left branch, a 6 V source with $R_2 = 3\ \Omega$ in the right branch (its polarity bucking the first source), and $R_3 = 6\ \Omega$ in the middle branch between junctions $a$ and $b$ . The strategy is the method verbatim: guess all three arrows, one junction equation, two loop walks, solve, then audit the answer with the one check that never lies — the power ledger. Sources are taken ideal here (fold any internal resistance into the branch resistors; EMF and Internal Resistance licenses exactly that move).

Step 1. Guesses: $I_1$ flows right along the top through $R_1$ toward $a$ ; $I_2$ leaves $a$ rightward through $R_2$ ; $I_3$ flows down the middle through $R_3$ . Junction $a$ : $I_1 = I_2 + I_3$ . (Junction $b$ would say the same — with $J = 2$ junctions, only $J-1 = 1$ statement is independent.)

Step 2. Loop 1, the left window, walked clockwise starting at $b$ : up through the 12 V source ( $+12$ ), along the top dropping through $R_1$ with the current ( $-4I_1$ ), down the middle with $I_3$ ( $-6I_3$ ), home. The loop rule: $12 - 4I_1 - 6I_3 = 0$ .

Step 3. Loop 2, the right window, walked clockwise starting at $b$ : up the middle against $I_3$ ( $+6I_3$ ), along the top with $I_2$ through $R_2$ ( $-3I_2$ ), then down through the 6 V source from its $+$ to its $-$ terminal ( $-6$ ), home: $6I_3 - 3I_2 - 6 = 0$ . Why the sign discipline is the whole game: every term's sign came from the crossing direction, not from intuition about which way charge "wants" to go. The method's power is that it needs no such intuition.

Step 4. Solve. From the junction equation substitute $I_1 = I_2 + I_3$ into loop 1: $12 - 4I_2 - 10I_3 = 0$ . Pair with loop 2 rearranged, $I_2 = 2I_3 - 2$ . Substituting: $12 - 4(2I_3 - 2) - 10I_3 = 0 \Rightarrow 20 - 18I_3 = 0$ , so

I_3 = \frac{10}{9} \approx 1.11\ \text{A}, \qquad I_2 = \frac{2}{9} \approx 0.22\ \text{A}, \qquad I_1 = \frac{4}{3} \approx 1.33\ \text{A}

All three came out positive, so all three guessed arrows happened to be right — the 6 V battery is being charged by the 12 V one? Check the crossing: $I_2$ enters the 6 V source at its $+$ terminal, so yes: charge is pushed backward through it, and this "battery" is absorbing energy. The rules report such facts without being asked.

Step 5. Audit with the power ledger. Delivered by the 12 V source: $12 \times \tfrac{4}{3} = 16.00$ W. Absorbed by the 6 V source: $6 \times \tfrac{2}{9} = 1.33$ W. Dissipated: $4I_1^2 + 3I_2^2 + 6I_3^2 = 7.11 + 0.15 + 7.41 = 14.67$ W. Ledger: $16.00 = 1.33 + 14.67$ . Exact. Verified: the linear system solved by matrix inversion gives (I1, I2, I3) = (1.3333, 0.2222, 1.1111) A and the power ledger balances to machine precision, 14.6667 W = 14.6667 W.

Go deeper · Derivation C · 2 minThe Wheatstone bridge — measuring resistance with a reading of zero

Derivation CThe Wheatstone bridge — measuring resistance with a reading of zero

We want the balance condition of the bridge in the second figure — four arms $R_1, R_2, R_3, R_x$ around a diamond, supply across one diagonal ( $e$ to $c$ ), galvanometer across the other ( $a$ to $d$ ) — and we want why this beats simply measuring $V/I$ across the unknown. Strategy: impose balance ( $I_G = 0$ ) and let the junction and loop rules collapse the circuit; then read off the null method's virtues. The subtlety to savour is that at balance the answer is independent of the supply voltage and of the galvanometer's own resistance — the two hardest things to know precisely in 1843.

Step 1. At balance no current crosses the galvanometer, so the junction rule at $a$ forces the current through $R_1$ to continue undiminished through $R_3$ — call it $I_{13}$ — and likewise one current $I_{2x}$ traverses $R_2$ then $R_x$ .

Step 2. Zero galvanometer current also means zero voltage across it (any real meter has finite resistance; $V_G = I_G R_G = 0$ regardless): points $a$ and $d$ sit at the same potential.

Step 3. Walk the upper loop $c\!-\!a\!-\!d\!-\!c$ with the loop rule: $-I_{13}R_1 + I_{2x}R_2 = 0$ , using $V_a = V_d$ to kill the middle term. Similarly the lower loop gives $I_{13}R_3 = I_{2x}R_x$ .

Step 4. Divide the two statements — the unknown currents cancel:

R_x = \frac{R_2 R_3}{R_1}

A pure ratio. With $R_1 = 100\ \Omega$ , $R_2 = 250\ \Omega$ and balance found at $R_3 = 40.0\ \Omega$ : $R_x = 250 \times 40/100 = 100.0\ \Omega$ . Why a null beats a deflection: the result contains no supply voltage (drift harmless), no galvanometer calibration (only its zero matters, and zeros are what galvanometers detect best — they are astonishingly sensitive near null), and precision comes from ratio arms, which can be manufactured and trimmed far more accurately than any absolute meter scale. Comparison against a standard is metrology's oldest, deepest trick.

Step 5. The same null philosophy, aimed at EMF instead of resistance, is the potentiometer: slide a tap along a uniform current-carrying wire until the galvanometer between tap and an unknown cell reads zero; at null the cell supplies no current, so its terminal voltage equals its true EMF — solving exactly the problem EMF and Internal Resistance raised, that any current drawn makes the terminals read $\varepsilon - Ir$ , not $\varepsilon$ . Poggendorff's compensation method (1841) measured chemistry honestly a century before high-impedance electronics existed. Verified: balance condition recomputed from the two loop equations; the numeric case gives R_x = 100.0 Ω; at balance the derivation holds for any galvanometer resistance, checked by re-solving the full five-branch network symbolically with R_G finite.

Limiting cases

case	result	meaning
circuit reducible by series/parallel	rules reproduce the shortcuts	equation (1) and (2) derive the reduction laws — see Series and Parallel Networks
one loop, one source	$I = \varepsilon/\sum R$	the single-loop formula of EMF and Internal Resistance is the loop rule alone
a guessed arrow is wrong	that current solves negative	flip the interpretation, not the algebra
bridge at balance	$I_G = 0$ , $R_x = R_2R_3/R_1$	supply and meter properties drop out
bridge off balance	five-branch network, no series/parallel reduction	the rules' full machinery is required — Derivation C's balance shortcut is the special case
changing flux threads the loop	$\sum \Delta V = -\,d\Phi_B/dt \neq 0$	loop rule's honesty clause: Faraday's territory
node that stores charge (capacitor plate)	$\sum I = dQ/dt$	junction rule generalises smoothly in RC Circuits

Common trapA negative answer or a wrong arrow means the analysis went wrong

Neither. The arrows you draw before solving are bookkeeping labels, not physical claims — the algebra is entirely indifferent to your guesses. A current that solves to $-0.7$ A is a perfectly correct answer: 0.7 A flowing opposite your arrow. The real errors live elsewhere: flipping a sign midway through a loop walk (decide the tour direction first, then never renegotiate), counting the same junction information twice (only $J-1$ node equations are independent), or choosing a "new" loop that revisits only used branches (each loop must contribute a branch no previous loop touched). Sign discipline in, correct physics out — the method is a machine precisely so that intuition, right or wrong, cannot derail it.

Worked examples

Example 1 — one loop, two opposing batteries. A 9 V battery and a 3 V battery are wired in one loop so their EMFs oppose, with $R_1 = 2\ \Omega$ and $R_2 = 4\ \Omega$ in the loop. Find the current and each battery's role.

Step 1. Walk the loop with the current arrow through the 9 V source from $-$ to $+$ : $+9 - 2I - 3 - 4I = 0$ . The 3 V term enters negative because the tour crosses it from $+$ to $-$ — its EMF bucks the walk.

Step 2. Solve: $6 = 6I$ , so $I =$ 1.0 A, positive — the 9 V battery sets the direction.

Step 3. Roles from the crossing directions: the 9 V battery delivers $9 \times 1 = 9$ W; the resistors burn $2(1)^2 + 4(1)^2 = 6$ W; the 3 V battery absorbs the remaining 3 W — it is being charged at 1 A. This two-battery loop is every charger you own: a supply EMF deliberately overpowering a cell's EMF through a current-limiting resistance.

Example 2 — the jump start, solved properly. A healthy battery ( $\varepsilon_1 = 12.6$ V, $r_1 = 0.02\ \Omega$ ) is clamped across a weak one ( $\varepsilon_2 = 11.4$ V, $r_2 = 0.15\ \Omega$ ) while the weak car's systems draw through an effective load $R = 0.5\ \Omega$ . Three branches meet at the shared terminals — find every current. (EMF and Internal Resistance treats the no-load version of mismatched pairing; here the rules handle the loaded case.)

Step 1. Call the common top-terminal potential $V$ (bottom terminals grounded, $0$ V) and let each branch current flow toward the top node from its source, and down through the load. The junction rule in node-voltage form: $\dfrac{12.6 - V}{0.02} + \dfrac{11.4 - V}{0.15} = \dfrac{V}{0.5}$ .

Step 2. Multiply out and solve: $V = 12.03$ V. (This "one unknown instead of three" organisation is the node-voltage method — the junction rule promoted to protagonist; it and its loop-side twin, mesh analysis, are the industrial forms of Kirchhoff.)

Step 3. Read off the branches: healthy battery supplies $(12.6 - 12.03)/0.02 =$ 28.3 A; the load takes $12.03/0.5 =$ 24.1 A; and the weak battery's branch gives $(11.4 - 12.03)/0.15 = -4.2$ A — negative: it is receiving 4.2 A of charge even while the car runs. Junction audit: $28.3 = 24.1 + 4.2$ , exact.

Step 4. The physics of the minus sign: the healthy battery holds the node at 12.03 V, above the weak cell's 11.4 V EMF, so current is forced backward through it — simultaneously running the car and charging the dead battery. One node equation predicted the whole choreography, including a current direction nobody guessed. Both examples recomputed numerically: Example 1 ledger 9 = 6 + 3 W exactly; Example 2 node voltage 12.034 V, branch currents 28.30 / 24.07 / −4.23 A, junction balance to three decimals.

Origins · 1 minGustav Kirchhoff and how this was discovered

Gustav Kirchhoff

1824–1887 · Königsberg, Breslau, Heidelberg, Berlin

Kirchhoff wrote these rules in 1845 as a 21-year-old physics student in Franz Neumann's Königsberg seminar, extending a seminar exercise on Ohm's law into the general laws of current branching. Two years later he went further than any circuit needed: his 1847 paper proved exactly how many independent equations a network of arbitrary shape supplies, inventing in passing the spanning-tree machinery that later generations recognised as founding results of graph theory — the count $B - J + 1$ used in this article is his. The rules were early work; the fame came at Heidelberg, where with Bunsen he founded spectroscopy, identified elements in the Sun, and stated the blackbody radiation law whose pursuit ultimately birthed quantum theory (that story is told at Scalar and Vector Diffraction Theory, whose integral also bears his name). But the student's laws have aged the best: every simulation of every chip on Earth is, at bottom, his two sentences applied a billion nodes at a time.

History

1827

— Ohm publishes the law of the single conductor; networks remain beyond reach

1833

— Christie exhibits the diamond null circuit at the Royal Society, to little notice

1841

— Poggendorff's compensation method measures EMF by balancing it against a known drop — null philosophy arrives

1843

— Wheatstone champions Christie's bridge, turning it into the precision resistance instrument that carries his name

1845

— Kirchhoff, aged 21, states the junction and loop rules in Neumann's Königsberg seminar

1847

— Kirchhoff's network paper counts the independent loops of any circuit, seeding graph theory

1873

— Maxwell's Treatise systematises mesh analysis: solve loops wholesale, by determinant

1958

— Kron and the network analysts carry Kirchhoff bookkeeping to power grids spanning continents

1973

— Nagel's SPICE at Berkeley automates nodal analysis; circuit simulation becomes universal engineering practice

Going further

References. Halliday, Resnick & Walker, Fundamentals of Physics, ch. 27 (multiloop circuits, ammeters and voltmeters); Purcell & Morin, Electricity and Magnetism §4.7–4.10; Griffiths, Introduction to Electrodynamics §7.1 for the field-level view of EMF and current steering; Kirchhoff's 1845 and 1847 papers (Annalen der Physik 72, and the network memoir) for the astonishing originals.

Provenance. Authored fresh for the SuperSTEM E&M track's circuits chapter (recon-§d gap heal); sources: v3 circuit pilots (rewritten to UG register) + legacy wiki circuit articles (narrative) — no AllQFT base exists for this topic.

Prerequisites: EMF and Internal Resistance · Current, Resistance, and Resistivity · Curl of the Electric Field (why potential is single-valued) · Continuity Equation (charge conservation at field level).

Leads to: Series and Parallel Networks (the reduction laws, derived from these rules) · RC Circuits (the junction rule meets charge storage) · Faraday's Law of Induction (the loop rule's honesty clause becomes physics).

How do I know which direction to draw the current arrows before solving?

You do not, and it does not matter. The arrows are bookkeeping labels that define what "positive" means for each branch; the physics is in the equations, not the guesses. Draw anything, keep the signs consistent during loop walks, and let the algebra report: positive means the arrow was right, negative means the flow is opposite. Experienced solvers guess plausibly to keep minus signs rare, but the answers are identical either way.

Why is one of the junction equations always redundant?

Because the whole circuit conserves charge globally. Add up the current-balance statements at every node: each branch current appears exactly twice, once leaving one node and once entering another, so the grand total is 0 = 0. Any one node's equation is therefore implied by the others — which is why a circuit with J junctions yields only J − 1 independent statements, and why using all J leads to a frustrating singular system.

Can the loop rule really fail? It is supposed to be energy conservation.

It is potential single-valuedness, which is narrower than energy conservation. Energy is always conserved; what fails when a changing magnetic flux threads your loop is the existence of a well-defined potential difference between two points — the induced electric field has circulation, so "the voltage from a to b" depends on the path taken. That is Faraday's law, not a paradox, and circuit theory absorbs it by packaging all changing flux inside inductors with terminal relation V = L dI/dt. Steady DC circuits are perfectly safe.

What does the galvanometer actually do in the bridge, and why must it read zero rather than something small?

A galvanometer is a sensitive current-to-needle transducer — microamps produce visible deflection. The bridge uses it purely as a null detector: you adjust R3 until the needle sits exactly on zero. At null, the unknown follows from the ratio R2R3/R1 with the supply voltage, the meter's calibration, and the meter's own resistance all absent from the result. A small nonzero reading would drag all three back into the answer. Detecting "zero versus not zero" is the easiest thing a meter does supremely well — that asymmetry is the entire null-method philosophy.

In Derivation B the 6 V battery absorbs energy. Is that physically real or an artefact of the sign conventions?

Entirely real — it is how every battery charger works. When a stronger source holds a battery's terminals above its own EMF, current is driven backward through it and the electrochemistry runs in reverse, storing energy (in rechargeable chemistries) or heating and gassing (in primary cells — do not try it). The rules flag the situation automatically: current crossing a source from + to − inside means the source is absorbing εI watts rather than delivering.

Do Kirchhoff's rules work for AC circuits too?

Yes, with upgraded bookkeeping. At any instant the junction and loop statements hold for the instantaneous currents and voltages of a lumped AC circuit; written for phasors, the same two rules govern complex amplitudes with impedances in place of resistances — that is exactly how AC Circuits and Phasors will analyse RLC networks. The caveats tighten with frequency: the lumped treaty (all flux in components, elements small compared with the wavelength) is what eventually expires, not the rules themselves.

How is this different from just using V = IR cleverly?

Ohm's law is a statement about one element; it cannot know how elements share current and voltage when wired together. Kirchhoff's rules are the sharing laws — they are what "wired together" means, mathematically. Series and parallel formulas are the rules pre-chewed for two special topologies (the next article derives them in three lines each); the moment a network leaves those topologies, you are back to the general machinery. Ohm supplies the branch relations; Kirchhoff supplies the network.