In simple terms
A friendly intro before the formal notes — no formulas yet.
Kirchhoff's laws
Cambridge 9702 Paper 2 — Kirchhoff's laws (10.2). Senpai Corner diagram-backed pilot with premium structure and live visuals.
- 1
Based on the principle of conservation of charge.
- 2
A 'junction' is any point where three or more conductors meet.
- 3
An alternative form is ΣI = 0, where currents entering are positive and currents leaving are negative.
- 4
Always label your assumed current directions on a diagram before starting your calculation.
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 10.2.1
Recall Kirchhoff's first law and understand that it is a consequence of conservation of charge
- 10.2.2
Recall Kirchhoff's second law and understand that it is a consequence of conservation of energy
- 10.2.3
Derive, using Kirchhoff's laws, a formula for the combined resistance of two or more resistors in series
- 10.2.4
Use the formula for the combined resistance of two or more resistors in series
- 10.2.5
Derive, using Kirchhoff's laws, a formula for the combined resistance of two or more resistors in parallel
- 10.2.6
Use the formula for the combined resistance of two or more resistors in parallel
- 10.2.7
Use Kirchhoff's laws to solve simple circuit problems
Explore the concept
Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.
Step-synced diagram — highlights what to look for in the simulation above.
Step 1
Based on the principle of conservation of charge.
13 more simulations for this topic — run them in the Simulations section below
Simulations
Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.
13 simulations · 5 to start with
Start herein this order — each one shows a different piece of the topic
- PhETStart here · 19702 9.1 · 10.2
Circuit Construction Kit: DC
Drag batteries, bulbs, resistors and meters onto a board; watch electrons flow.
Why this one: Put an ammeter in each parallel branch and add them: current in equals current out at a junction.
Try this
- Build battery → bulb → back; show “Electrons” and the ammeter — read I.
- Add a second bulb in parallel and put an ammeter in each branch — sum the currents.
- Place the ammeter before and after a bulb — is the current the same?
Look for Current is the same all along a series loop and splits at a junction (Σ I in = Σ I out).
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- 3JCN PhysicsStart here · 29702 10.1 · 9702 10.2 · IB B.5
Resistors in Parallel
Resistors in parallel: see voltage shared and current divided
Why this one: Branch currents sum to the supply current: Kirchhoff's first law with meters.
Try this
- Read the voltage across each parallel resistor.
- Read the branch currents and sum them.
Look for Each parallel branch sees the full voltage and the branch currents add to the total.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsStart here · 39702 10.1 · 9702 10.2 · IB B.5
Resistors in Series
Resistors in series: see current shared and voltage divided
Why this one: Voltages across series resistors add to the e.m.f.: Kirchhoff's second law round one loop.
Try this
- Read the current at each point in the series circuit.
- Read the voltage across each resistor and sum them.
Look for The same current flows through every series resistor and the voltages add to the supply.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- The Physics ClassroomStart here · 49702 10.2 · IB B.5
Equivalent Resistance
Given a series, parallel or combination circuit, choose resistor values that produce a target equivalent resistance
Why this one: Choose resistor values to hit a target: series add, parallel combine by reciprocals.
Try this
- Solve a series target.
- Solve a parallel target.
- Try a combination circuit.
Look for Series resistances add, and parallel resistances combine by reciprocals.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- oPhysicsStart here · 59702 10.1 · 9702 10.2 · IB B.5
Electric Circuit with Four Identical Lightbulbs
Combination circuit with four identical bulbs and three switches; predict then test brightness and current
Why this one: Closing a parallel branch draws extra current from the source; predict the brightness changes.
Try this
- Close one switch at a time and compare bulb brightness.
- Close all three switches.
- Open the switches in a different order.
Look for Bulbs in series share the current and glow dimmer; a parallel branch draws extra current from the source.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
More simulations8 more on this topic — core ones first
- The Physics ClassroomCore9702 9.1 · 9702 9.2 · 9702 10.1
DC Circuit Builder
A virtual circuit board: add resistors, bulbs, wires and ammeters, use a voltmeter, and build series, parallel and combination circuits
Try this
- Build a series circuit and read the ammeter.
- Rebuild the same resistors in parallel and compare.
- Measure the voltage across each resistor with the voltmeter.
Look for Current is the same everywhere in series, and voltage is the same across parallel branches.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- SimuPhysicsCore9702 10.1 · 9702 10.2 · IB B.5
Parallel Connections: Wires and Nodes
Eight drawings of parallel resistors: gather every starting end at node a and every finishing end at node b and check each branch
Try this
- Pick a drawing and gather the ends at node a and node b.
- Check each branch connects to both nodes.
- Try the Ladder and Diamond drawings.
Look for Resistors are in parallel when every branch connects the same two nodes.
Open on SimuPhysicsRuns on their siteSimuPhysics by Mohamed Abdelsalam · Licensed to MarkScheme (site publishes no licence; served with frame-ancestors self)
- 3JCN PhysicsCore9702 9.2 · 9702 9.3 · 9702 10.1
Ohm's Law - Resistors in Parallel
Ohm's law applied to resistors in parallel with meters
Try this
- Read each branch ammeter and the voltmeter.
- Check V = IR for each branch.
- Sum the branch currents and compare with the total.
Look for All branches share one voltage and the branch currents sum to the supply current.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 9.2 · 9702 9.3 · 9702 10.1
Ohm's Law - Resistors in Series
Ohm's law applied to resistors in series with meters
Try this
- Read the ammeter and each voltmeter.
- Check V = IR for each resistor.
- Check that the voltmeter readings sum to the supply voltage.
Look for One current, and the sum of the resistor voltages equals the emf.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 10.1 · 9702 10.2 · IB B.5
DC Circuit Construction Kit
Construct DC circuits with batteries, bulbs and meters
Try this
- Connect one battery and one bulb and add an ammeter.
- Add a second bulb in series and compare the ammeter reading.
- Move the second bulb to parallel and compare.
Look for Series bulbs share the current and dim; parallel bulbs each get the full voltage.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 10.1 · 9702 10.2 · IB B.5
Resistor Combination Construction Kit
Build resistor combinations and compute the equivalent resistance
Try this
- Build two equal resistors in series and compute the equivalent.
- Rebuild them in parallel and compare.
Look for Series resistances add; parallel resistances combine to less than the smallest.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
Key formulas
Tap any symbol to reveal exactly what it means and its units.
Tap a symbol — great for exam definitions
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Full topic notes
Formal explanation with the rigour you need for the exam.
Kirchhoff's First Law: The Current Rule (KCL)
Imagine a crossroads for electrical current. Kirchhoff's First Law, often called the Current Law (KCL), tells us that the total amount of electric charge flowing into any junction in a circuit must exactly equal the total amount of charge flowing out of it. This is a direct consequence of the principle that charge is conserved – it can't accumulate or disappear at a point.
Based on the principle of conservation of charge.
A 'junction' is any point where three or more conductors meet.
An alternative form is ΣI = 0, where currents entering are positive and currents leaving are negative.
Always label your assumed current directions on a diagram before starting your calculation.
Kirchhoff's Second Law: The Voltage Rule (KVL)
Now, consider a complete journey around any closed loop within a circuit. Kirchhoff's Second Law, the Voltage Law (KVL), states that if you add up all the potential differences (voltages) encountered along this loop, the algebraic sum must be zero. This law directly stems from the conservation of energy; no energy is gained or lost when you return to your starting point. It can also be stated as: the sum of the e.m.f.s in a closed loop equals the sum of the potential drops.
Sign Conventions for Applying KVL
Correctly applying KVL depends on a consistent sign convention. This is a common source of errors, so follow these steps carefully:
1. Assume Current Directions: First, draw arrows on your circuit diagram for the direction you think the current flows in each branch. If you guess wrong, the final answer for that current will simply be negative.
2. Choose a Loop Direction: Decide on a direction to trace each loop (e.g., clockwise). This choice is arbitrary but must be kept consistent for the entire loop.
3. Apply the Rules: As you trace your loop:
- e.m.f. sources (Cells): If you move from the negative to the positive terminal, the e.m.f. is positive (a potential rise). If you move from positive to negative, the e.m.f. is negative.
- Resistors: If you move through a resistor in the same direction as your assumed current, the potential difference (IR) is negative (a potential drop). If you move against the current, the p.d. is positive.
Applying KVL: Essential Concepts
To apply Kirchhoff's Second Law effectively, you need to be familiar with a few key concepts. The electromotive force (e.m.f., \epsilon) is the total energy supplied by a source per unit charge. However, power sources aren't perfect; they have internal resistance () which dissipates some energy internally.
When current flows, a potential drop occurs across this internal resistance, known as lost volts (). The actual voltage delivered to the external circuit is the terminal potential difference (). The relationship is straightforward: the total e.m.f. is the sum of the terminal potential difference and the lost volts.
When applying KVL, always assume a direction for your loop traversal. If you move through a component in the assumed direction of current and it's a resistor, it's a potential drop (negative). If you move from negative to positive terminal of a cell, it's a potential rise (positive e.m.f.). If your calculated current turns out negative, it simply means your initial assumed direction was opposite to the actual flow!
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
A circuit consists of a 6V battery (negligible internal resistance) and two resistors, R1 = 3\Omega and R2 = 6\Omega. R1 and R2 are connected in parallel. This parallel combination is then connected in series with a third resistor R3 = 2\Omega. Find the total current drawn from the battery using Kirchhoff's Laws.
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Identify Junctions and Label Currents: Let the current leaving the battery be I_total. This current splits into I1 through R1 and I2 through R2 at the first junction. They recombine to form I_total again before going through R3.
A circuit contains two cells. Cell A has an e.m.f. of 9.0 V and an internal resistance of 1.0 Ω. Cell B has an e.m.f. of 3.0 V and an internal resistance of 0.5 Ω. They are connected in a loop with a 5.0 Ω resistor. The positive terminal of Cell A is connected to the positive terminal of Cell B, so they oppose each other. Calculate the current in the circuit and the terminal potential difference across Cell A.
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Draw the Circuit and Assume Current Direction: Let's assume the current flows clockwise, driven by the larger e.m.f. of Cell A. The current will flow out of the positive terminal of Cell A, through the 5.0 Ω resistor, and into the positive terminal of Cell B.
How it all connects
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Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Key terms for this topic — skim now; the Check step will test them.
- charge is conserved
It can't accumulate or disappear at a point.
- conservation of energy
This law directly stems from the conservation of energy; no energy is gained or lost when you return to your starting point. It can also be stated as: the sum of the e.m.f.s in a closed loop equals the sum of the potential drops.
- electromotive force (e.m.f., \epsilon)
The electromotive force (e.m.f., \epsilon) is the total energy supplied by a source per unit charge.
- Write the formula
\epsilon = V + v
- Junction
A point in an electrical circuit where three or more conductors meet, allowing current to split or combine.
- mathematical
\Sigma I_{in} = \Sigma I_{out}
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
If you can explain it simply, you own it — gaps here are marks you’d lose.
Teach it back
Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.
Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
Based on the principle of conservation of charge.
A 'junction' is any point where three or more conductors meet.
An alternative form is ΣI = 0, where currents entering are positive and currents leaving are negative.
Always label your assumed current directions on a diagram before starting your calculation.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
the p.d. measured by the voltmeter.
Use Kirchhoff's laws to show that the total resistance R_T of the external circuit is given by 1/R_T = 1/R₁ + 1/R₂.
Extra simulations & links
PhET, GeoGebra and other curated tools — open in a new tab.
Frequently asked
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do 9702/23 · Q5(c)(ii) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
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