Kirchhoff's laws

A-Level Physics· Unit 10: D.C. circuits· 40 min read

1. Kirchhoff's First Law (Junction Rule)★★☆☆☆⏱ 15 min

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📘 Definition

Kirchhoff's First Law

\sum I_{in} = \sum I_{out}

The sum of currents entering a junction equals the sum of currents leaving the junction. This is a direct consequence of conservation of charge: charge cannot be created or destroyed at a junction.

Example:

If 3 A and 2 A enter a junction, 5 A must leave it.

When analysing circuits, first label all currents at every junction with an assumed direction. If you pick the wrong direction, the calculated current will be negative, indicating the actual direction is opposite to your assumption.

📐 Worked Example

At a junction, current $I_{1} = 2.0$ A enters, and $I_{2} = 1.5$ A leaves. Find the magnitude and direction of the third current $I_{3}$ .

1
Apply KCL, taking currents entering the junction as positive:
2
$I_1 = I_2 + I_3$
3
Rearrange to solve for $I_{3}$ :
4
$I_3 = I_1 - I_2 = 2.0 - 1.5 = 0.5 \ \text{A}$
5
Since $I_{3}$ is positive, our assumption that it leaves the junction is correct. The third current is 0.5 A leaving the junction.

Exam tip:

Always state the direction of the final current to earn full marks in structured questions.

2. Kirchhoff's Second Law (Loop Rule)★★★☆☆⏱ 20 min

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📘 Definition

Kirchhoff's Second Law

\sum \varepsilon = \sum IR

Around any closed loop in a circuit, the sum of electromotive forces (emfs) equals the sum of potential differences across all resistors in the loop. This is a consequence of conservation of energy.

Example:

For a single loop with one battery and one resistor, this simplifies to $ε = I R$ .

Sign convention is critical here: when traversing the loop, assign a positive value to emf if you move from the negative to positive terminal of a battery. Assign a negative potential difference $- I R$ if you move across a resistor in the same direction as the current.

📐 Worked Example

A single loop contains a 6.0 V battery and two series resistors of $2.0 Ω$ and $4.0 Ω$ . Find the current in the loop using KVL.

1
Traverse the loop clockwise starting at the negative terminal of the battery.
2
Apply KVL: sum of emfs equals sum of IR drops:
3
$\varepsilon = I R_1 + I R_2$
4
Substitute the given values:
5
$6.0 = I (2.0 + 4.0) = 6I$
6
Solve for I:
7
$I = 1.0 \ \text{A}$

3. Solving Complex Two-Loop Circuits★★★★☆⏱ 25 min

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Most CIE exam questions requiring Kirchhoff's laws involve two connected loops with multiple junctions. To solve these, you write a system of simultaneous equations from KCL and KVL, then solve for unknown currents.

Label all currents and choose their assumed directions at every junction
Write one independent KCL equation for each junction
Write one independent KVL equation for each closed loop
Solve the system of simultaneous equations
Interpret negative values to find actual current directions

📐 Worked Example

Two loops share a $2 Ω$ resistor. Loop 1 has a 12 V battery and $1 Ω$ resistor. Loop 2 has a 6 V battery and $3 Ω$ resistor. Find the current in the shared resistor.

1
Label currents: $I_{1}$ in loop 1, $I_{2}$ in loop 2. By KCL: total current through shared resistor is $I = I_{1} + I_{2}$ :
2
Write KVL for loop 1:
3
$12 = 1I_1 + 2(I_1 + I_2) = 3I_1 + 2I_2$
4
Write KVL for loop 2:
5
$6 = 3I_2 + 2(I_1 + I_2) = 2I_1 + 5I_2$
6
Solve simultaneous equations: multiply first by 2, second by 3, subtract to get $I_{2} = - 6/11 \approx - 0.55$ A, then $I_{1} \approx 4.36$ A:
7
Total current in shared resistor: $I = I_{1} + I_{2} \approx 3.8$ A. Negative $I_{2}$ means its direction is opposite to our assumption.

4. Common Pitfalls

Wrong move:

Mixing up the physical basis of the two laws

Why:

This common mix-up costs easy marks when examiners ask for an explanation of the laws

Correct move:

Remember: KCL = conservation of charge, KVL = conservation of energy

Wrong move:

Ignoring negative current values

Why:

Candidates often discard negative values instead of interpreting them correctly

Correct move:

A negative value just means the actual current direction is opposite to your assumed direction

Wrong move:

Inconsistent sign conventions for loop traversal

Why:

Randomly assigning signs leads to incorrect final values

Correct move:

Always use the same convention: +emf for negative to positive, -IR for same direction as current

Wrong move:

Writing more independent equations than needed

Why:

Extra equations are redundant and can introduce errors when solving

Correct move:

For n independent loops, you need n total equations (n-1 KCL, n KVL)

5. Quick Reference Cheatsheet

Law	Mathematical Form	Physical Basis	Key Tip
First (Junction)	$\sum I_{in} = \sum I_{o u t}$	Conservation of charge	Negative = opposite direction
Second (Loop)	$\sum ε = \sum I R$	Conservation of energy	Stick to consistent signs

6. Frequently Asked

Do I need to remember why Kirchhoff's laws work?

Yes, CIE examiners regularly ask to link KCL to conservation of charge and KVL to conservation of energy, so you must be able to explain this connection.

Which direction should I choose for currents?

You can choose any direction. A negative final value just means the actual current flows opposite to your chosen direction.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 12
Solve two-loop junction problem
2023 · 22
Derive current from KCL/KVL
2021 · 11
Explain physical basis of laws

Going deeper

syllabusOfficial CIE 9702 SyllabusUnit 10 reference

What's Next

Kirchhoff's laws are the foundation of all circuit analysis, and they underpin every topic you will cover in DC and later AC circuits. Now that you understand how to apply these rules, you can move on to more practical circuit concepts including internal resistance of batteries and potential dividers, which are extremely common exam topics that build directly on this foundation. Mastery of Kirchhoff's laws also prepares you for more advanced circuit analysis in A2 Physics, including AC impedance and complex network problems.