Why two more rules are needed
Electric circuits generally consist of a number of resistors and cells interconnected, sometimes in a complicated way. The formulae derived earlier for series and parallel combinations of resistors are not always sufficient to determine all the currents and potential differences in the circuit. Two rules, called Kirchhoff's rules, are very useful for the analysis of electric circuits.
Before stating them, we adopt a labelling habit that removes most of the confusion students meet. Given a circuit, we begin by labelling the current in each resistor by a symbol, say $I$, together with a directed arrow showing the direction in which $I$ is taken to flow. If $I$ ultimately turns out positive, the actual current is in the direction of the arrow. If $I$ turns out negative, the current actually flows opposite to the arrow. The arrows are therefore guesses — the algebra corrects them. The same is done for each source: its positive and negative electrodes are marked, along with an arrow for the current through it.
For a cell, this labelling fixes the potential difference between the positive terminal $P$ and negative terminal $N$. If the current $I$ flows from $N$ to $P$ inside the cell, then $V = V(P) - V(N) = \varepsilon - Ir$. If, while labelling, one instead goes from $P$ to $N$, the term reverses and $V = \varepsilon + Ir$. This single convention quietly absorbs the internal resistance into the same accounting as every external resistor, so there is nothing extra to remember when a cell is non-ideal.
Why two rules and not one? Each captures a different conserved quantity. The junction rule is a statement about where charge goes — it cannot pile up at a node — and so it relates currents. The loop rule is a statement about energy per unit charge — a charge ferried around a closed path returns to the same potential — and so it relates EMFs and potential drops. Between them they generate exactly enough equations to pin down every unknown current in even a tangled mesh, which is why no separate "third rule" is ever needed.
The junction rule (KCL)
Junction rule: At any junction, the sum of the currents entering the junction is equal to the sum of the currents leaving the junction. This applies equally well if, instead of a junction of several lines, we consider a point lying within a single line.
The proof follows from the fact that when currents are steady, there is no accumulation of charge at any junction or at any point in a line. The total current flowing in — which is the rate at which charge flows into the junction — must therefore equal the total current flowing out. The junction rule is, in this exact sense, a statement of conservation of electric charge. Bending or reorienting the wire does not change its validity.
Written compactly, with currents into a junction taken positive and currents out taken negative, the rule reads $\sum I = 0$. For the figure above, $I_1 + I_2 = I_3$, exactly the relation a balanced bank account would obey.
Which rule is "charge" and which is "energy"?
Assertion–reason and statement questions routinely swap the two conservation laws. The junction rule is conservation of charge (nothing piles up at a node); the loop rule is conservation of energy (potential returns to its starting value around a loop). Mixing these up is the single most common error in this topic.
Junction = charge conservation. Loop = energy conservation. Never the reverse.
The loop rule (KVL)
Loop rule: The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero.
This rule is also obvious once stated, since the electric potential depends only on the location of the point. Starting at any point and coming back to that same point, the total change in potential must be zero. In a closed loop we do come back to the starting point, and hence the rule. Because potential energy per unit charge is what we are tallying, the loop rule is a statement of conservation of energy: whatever energy a unit charge gains crossing the EMF sources, it loses again as $IR$ drops across the resistances by the time it returns home.
Sign conventions for a loop
The loop rule is only as reliable as the bookkeeping behind it. Fix a direction in which you will walk around the loop — say clockwise — and apply the same two conventions at every element you cross. The choice of walking direction is free; what matters is consistency.
| Element crossed | Condition while traversing | Potential change |
|---|---|---|
| Resistor $R$ | traversed along the assumed current | − I R (a drop) |
| Resistor $R$ | traversed against the assumed current | + I R (a rise) |
| Cell (EMF $\varepsilon$) | entered at $-$ and left at $+$ (i.e. $-\!\to\!+$) | + ε |
| Cell (EMF $\varepsilon$) | entered at $+$ and left at $-$ (i.e. $+\!\to\!-$) | − ε |
| Internal resistance $r$ | treated exactly like any resistor | ∓ I r |
Read the table once more in plain words: the EMF of a source enters with a positive sign when you traverse it from its negative to its positive terminal, and the $IR$ term enters with a negative sign whenever you move in the direction of the current through that resistor. Sum every such contribution around the closed loop and set the total to zero.
EMF sign vs IR-drop sign
The sign of a cell's EMF depends on which terminal you enter first — not on the current direction. The sign of an $IR$ term depends on the current direction relative to your traversal — not on the cell. Students who tie the two together (for example, always making EMF positive whenever current flows out of $+$) get systematically wrong equations. Decide each contribution independently using the table.
EMF sign ← order of terminals crossed. IR sign ← current direction vs traversal direction.
Step-by-step method for a network
For circuits not reducible to series and parallel — a multi-loop mesh, a Wheatstone bridge, a bridged ladder — the following routine never fails. The aim is always to produce exactly as many independent equations as there are unknown currents.
| Step | What to do | Rule used |
|---|---|---|
| 1 | Assign an arrow and a symbol to the current in every branch. Direction is a free guess. | Labelling |
| 2 | At each junction, write currents in terms of as few unknowns as possible to cut down the count. | Junction rule |
| 3 | Choose enough independent closed loops to match the remaining unknowns; fix one traversal direction per loop. | Loop rule |
| 4 | Write the algebraic sum of potential changes for each loop, set to zero, using the sign table. | Loop rule |
| 5 | Solve the simultaneous equations. A negative answer means that branch's real current opposes its arrow. | Algebra |
Assume a direction — let the sign decide
Many students freeze at the start, trying to guess the "true" current direction in every branch. You do not have to. Pick any direction, solve, and read the sign. A positive result confirms your guess; a negative result tells you the current really flows the other way, with the magnitude unchanged. Re-drawing the circuit after a negative answer is wasted effort.
Direction is a hypothesis; the sign of $I$ is the verdict. Trust the algebra.
The cleanest application of both rules together is the Wheatstone bridge — junction rule at the bridge nodes, loop rule around two meshes, and the balance condition falls out directly.
Worked two-loop network
The power of the rules shows when symmetry cannot rescue you. The NCERT network of Example 3.6 has three unknown branch currents and is solved by applying the loop rule to three closed loops after the junction rule has fixed the branch currents in terms of $I_1$, $I_2$ and $I_3$.
Determine the current in each branch of the network. Two cells (10 V and 5 V) and several $2\,\Omega$, $4\,\Omega$, $1\,\Omega$ resistors form a two-loop mesh; the branch currents have been reduced to three unknowns $I_1$, $I_2$, $I_3$ using the junction rule.
Loop ADCA (second rule):
$10 - 4(I_1 - I_2) + 2(I_2 + I_3 - I_1) - I_1 = 0 \;\Rightarrow\; 7I_1 - 6I_2 - 2I_3 = 10$
Loop ABCA:
$10 - 4I_2 - 2(I_2 + I_3) - I_1 = 0 \;\Rightarrow\; I_1 + 6I_2 + 2I_3 = 10$
Loop BCDEB:
$5 - 2(I_2 + I_3) - 2(I_2 + I_3 - I_1) = 0 \;\Rightarrow\; 2I_1 - 4I_2 - 4I_3 = -5$
These three simultaneous equations solve to give
$I_1 = 2.5\ \text{A}, \quad I_2 = \tfrac{5}{8}\ \text{A}, \quad I_3 = \tfrac{1}{8}\ \text{A}.$
It is easily verified that applying the loop rule to any remaining closed loop yields no new independent equation — the values above already satisfy the loop rule for every loop of the network. That self-consistency is the practical check on your answer.
Two structural lessons sit inside this example. First, the junction rule did the heavy lifting at the outset, collapsing six branch currents to three unknowns. Second, three loops sufficed because there were three unknowns; a fourth loop equation would be redundant. This matching of "independent loops to unknowns" is what keeps Kirchhoff problems finite and solvable.
Kirchhoff's Rules in one screen
- Junction rule (KCL): currents in = currents out at any node; $\sum I = 0$. It is conservation of charge for steady currents.
- Loop rule (KVL): algebraic sum of potential changes around any closed loop = 0. It is conservation of energy.
- Sign of $IR$: $-IR$ when you traverse a resistor along the current, $+IR$ against it.
- Sign of EMF: $+\varepsilon$ entering at $-$ and leaving at $+$; $-\varepsilon$ the other way. Internal resistance behaves like any resistor.
- Method: label arrows, reduce unknowns by the junction rule, write one loop equation per remaining unknown, solve. A negative current just means the real direction is reversed.
- When to use: any network not reducible to plain series and parallel — the Wheatstone bridge being the classic application.