Maxwell's Equations
All four equations visualized. Builds on Gauss's Law and Faraday's Law.
James Clerk Maxwell noticed a problem with Ampère’s law. Consider a charging capacitor: current flows in the wire toward one plate and away from the other. If you draw an Ampèrian loop around the wire, you detect current. But if you stretch your surface to pass BETWEEN the plates (where no wire exists), there’s no current — yet the loop is the same.
How did Maxwell fix this contradiction?
Maxwell added the displacement current! Between the capacitor plates, there’s no physical current, but the electric field is changing as the capacitor charges. Maxwell proposed that this changing E-field acts as a source of magnetic field, just like a real current. He called it “displacement current”: I_d = ε₀(dΦ_E/dt). This single addition completed the equations of electromagnetism and predicted that light is an electromagnetic wave!
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Cycle through all four of Maxwell’s equations with animated visualizations. Watch Gauss’s law show charges radiating field lines, Faraday’s law show a changing B creating E, and the Ampère-Maxwell law show displacement current between capacitor plates. The finale: all four equations together predict a self-sustaining electromagnetic wave!
∇×B = μ₀J + μ₀ε₀(∂E/∂t)
This is the Ampère-Maxwell law — the fixed version. The second term is Maxwell’s addition: a changing electric field creates a magnetic field. Combined with Faraday’s law (changing B creates E), this predicts self-sustaining electromagnetic waves.
Every wireless signal is a Maxwell’s equation prediction come true. An oscillating current in an antenna creates a changing E-field, which creates a changing B-field, which creates a changing E-field... and the wave propagates through space at the speed of light.
Light is an electromagnetic wave — oscillating E and B fields sustaining each other through empty space. Maxwell’s equations predicted this. The sun’s light takes 8 minutes to reach Earth, traveling at c = 1/√(μ₀ε₀).
The oldest light in the universe (from 380,000 years after the Big Bang) is an electromagnetic wave that has been traveling for 13.8 billion years. It obeys Maxwell’s equations perfectly.
Capacitive touchscreens rely on the same displacement current concept Maxwell introduced. Your finger changes the electric field at the screen’s capacitive sensor, creating a detectable signal — no physical current needed.
“Four equations. That’s it. Four equations describe every electric and magnetic phenomenon in the classical universe. They predicted light, radio, X-rays, and gamma rays — all before any of those waves were deliberately generated. Einstein called Maxwell’s work “the most profound and the most fruitful that physics has experienced since the time of Newton.””
Maxwell’s Equations Lab
Explore all four of Maxwell’s equations interactively. Toggle each equation on or off to see how removing one breaks the physics. When all four are active, watch an electromagnetic wave emerge and propagate across the canvas.
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Start with all four equations active and watch the EM wave propagate. Now disable Faraday’s law — the wave dies because changing B can no longer create E. Re-enable it and disable the displacement current term — the wave dies again because changing E can no longer create B. Both are needed for light to exist! Try adjusting the frequency and watch the wavelength change. Red light has a longer wavelength than blue. All traveling at the same speed: c.
Maxwell’s equations show that electricity, magnetism, and light are all manifestations of a single phenomenon: the electromagnetic field. The displacement current term was the key — it completed the symmetry between E and B and predicted that electromagnetic disturbances propagate at the speed of light. Maxwell’s equations are arguably the most important equations in all of physics.