Kirchhoff's Rules: Multi-Loop Circuit Lab
Change the EMFs and resistors in a real two-loop circuit and watch all three branch currents solve live from Kirchhoff's voltage and current laws.
This two-loop circuit has two batteries and three resistors sharing a middle branch. Change any value and watch all three branch currents update live, solved from Kirchhoff's voltage and current laws.
About the Kirchhoff's Rules: Multi-Loop Circuit Lab
Free kirchhoff's rules: multi-loop circuit lab. Change the EMFs and resistors in a real two-loop circuit and watch all three branch currents solve live from Kirchhoff's voltage and current laws. Drag, change the sliders and see the result live. No sign-up, works on phone and computer. Built for physics, the kirchhoff's rules: multi-loop circuit lab runs instantly in your browser: change a setting or drag an object and the result updates at once, so you learn by trying things out rather than only reading about them.
Change the EMFs and resistors in a real two-loop circuit and watch all three branch currents solve live from Kirchhoff's voltage and current laws. Use it to explore physics ideas at your own pace, then check what you found against the key ideas further down this page.
How to use the Kirchhoff's Rules: Multi-Loop Circuit Lab
- Use the controls to change EMF ε₁ (left battery), EMF ε₂ (right battery), Resistor R₁ (left branch), Resistor R₂ (right branch), Resistor R₃ (middle branch). The simulation reacts instantly.
- Press "Reset to defaults", "Lab Report" to start, reset or change what is happening.
- Where you see a glowing handle, object, weight or atom, drag it with your mouse or finger. Everything responds in real time.
- Watch the readouts and graphs update as you experiment, and compare what you see with the key ideas below.
Things to try
- Set both EMFs equal and see how the middle-branch current behaves.
- Raise R3 (the shared branch) very high and see the loops nearly decouple.
- Try the middle-branch current challenge.
- Verify the junction rule holds for several different slider settings.
Key ideas you can learn
- Kirchhoff's voltage law (KVL): the sum of EMFs and voltage drops around any closed loop is zero.
- Kirchhoff's current law (KCL): the total current flowing into a junction equals the total current flowing out.
- A circuit with more than one loop generally needs both laws together, solved as a system of simultaneous equations, to find every branch current.
- Mesh analysis - assigning a circulating current to each loop and writing one KVL equation per loop - is a systematic way to solve any multi-loop resistor network.
Show my work
Challenges
Challenge 1 - solve for the middle-branch current
Using the current slider values, write the two loop (KVL) equations and the one junction (KCL) equation, solve the 2×2 system by hand, and enter your value for I₃ (the middle-branch current) in amps.
Challenge 2 - junction rule check
Kirchhoff's current law says the current into a junction equals the current out. With I₁ defined flowing into the middle node and I₂, I₃ flowing out, I₁ should equal I₂ + I₃. Read off the sim's current values and enter I₁ − I₂ − I₃ (it should be extremely close to zero).
Where this is used in the real world
Electrical engineers use Kirchhoff's rules and mesh analysis exactly like this to design and troubleshoot multi-branch circuits in everything from car wiring to power distribution networks.
Who is this simulation for?
Physics students in middle school, high school and first-year university, teachers who want a quick demonstration for the projector, and anyone revising for exams. It works well for flipped classrooms because students can explore before the lesson.
For teachers: project it on the board, let students predict what will happen, then run it together. For students: change one thing at a time and write down what changes.
Frequently asked questions
Why can't Ohm's law alone solve a two-loop circuit?
Ohm's law only relates voltage, current and resistance for a single resistor; a multi-loop circuit has multiple unknown currents linked by shared branches, so you need Kirchhoff's two laws together to get enough independent equations.
What does it mean if the junction check (I1 - I2 - I3) isn't exactly zero?
In a real solved circuit it should be extremely close to zero; a large mismatch would mean the current directions or the mesh equations were set up incorrectly.
Is the Kirchhoff's Rules: Multi-Loop Circuit Lab free to use?
Yes. It is completely free, with no signup, no download and no ads inside the simulation. It runs in your web browser.
Does the Kirchhoff's Rules: Multi-Loop Circuit Lab work on a phone or tablet?
Yes. It uses touch as well as the mouse, so you can drag objects with your finger. A larger screen makes the controls easier to see.