🔌 Faraday's law of induction

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🎯 Visualization of the relationship between a changing magnetic field and the induced electric field
🧠 Inspired by original version on web page of Rob Salgado
🐍 A VPython demo is available as well, see faradays_law.py
👉 Current through a real wire would have an azimuthal magnetic around the wire
👉 Faraday’s law

$\text{In differential form:}\quad \nabla \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t}$

$\text{In integral form:}\quad \oint \overrightarrow{E}\cdot d\overrightarrow{l} = -\frac{d\Phi_B}{dt}$

🔴 Arrows represent the magnetic field change $\frac{\partial \overrightarrow{B}}{\partial t}$
🔵 Arrows represent induced electric field $\nabla \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t}$
🟡 Balls visualize the electric charges traveling through the wire.
🟢 Faraday loops represent the orientation of the integration loop used in Faraday’s law.

These green arrows do not necessarily represent the physical electric field direction itself. Instead, they define the positive circulation direction according to the right-hand rule:

Live experiment

Important conceptual points

The simulation distinguishes between:

Visual Element	Meaning
Green loop direction	chosen mathematical orientation
Blue electric field	physical induced field
Red magnetic arrows	changing magnetic flux source

This separation is important because Faraday’s law relates:

• the circulation of the electric field around a closed path,
• to the time rate of change of magnetic flux through that path.

Physical Interpretation of the Minus Sign

The minus sign in Faraday’s law means:

Nature resists changes in magnetic flux.

If the magnetic field through the loop increases in one direction, the induced electric field circulates in the direction that would generate an opposing magnetic field. This is the essence of electromagnetic induction and energy conservation.

Field of moving charged rod

About

🎯 Illustration of Faraday’s law: field around a moving charged rod
🧠 Inspired by MovingLineOfCharge by Ksenja Llazar and Alyssa McCaskey
🐍 A VPython demo is available as well, see charged_rod.py
👉 The rod is modeled as a collection of point charges
👉 Electric field arrows are red, magnetic field arrows are blue

Live experiment