The Maxwell equations

Equipment

• “Model A” / “Model B” buttons switching between field geometry and the EM wave
• “Amplitude / source” and “Wavelength / scale” sliders
• “Frequency / speed” and “View angle / coefficient” sliders
• Two rotatable 3D figures and a “Pause” button

Procedure

1. Read the field geometry of the four equations

   In “Model A”, drag to rotate and vary “Amplitude / source”: identify field lines starting/ending on charges ($\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$) and the closed loops of the magnetic field ($\nabla\cdot\mathbf{B} = 0$). Orange and teal distinguish the two vector senses.

2. Watch the self-sustaining electromagnetic wave

   Switch to “Model B” and press “Pause” at an instant: check on the figure that $\mathbf{E}$ and $\mathbf{B}$ are mutually perpendicular, in phase, and both perpendicular to the propagation direction — exactly the plane-wave solution of the Faraday and Ampère–Maxwell equations.

3. Check c = λf and the viewing angle

   Raise “Frequency / speed” and see that “Wavelength / scale” must shrink to keep $c = \lambda f$; use “View angle / coefficient” to turn the wave and inspect the polarization direction. Infer the role of $\varepsilon_0\mu_0$: Maxwell predicted $c = 1/\sqrt{\varepsilon_0\mu_0}$, matching the speed of light.

Simulation

Experiment history

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