The four equations unified, displacement current, and symmetry
Four equations hold all of classical electricity and magnetism. They say how charges make fields, how fields make each other, and why light exists.
Click each equation to see what it says in pictures: charges source E, B has no sources, changing B makes E, and currents plus changing E make B.
Click any equation card to see an animated visual interpretation.
Maxwell's four equations unify all of electricity and magnetism into a single elegant framework. They describe how charges create fields, how fields create each other, and predict the existence of electromagnetic waves traveling at the speed of light.
Click each equation to see its visual interpretation. Together, they tell a complete story: charges source electric fields, there are no magnetic monopoles, changing B induces E, and currents plus changing E induce B.
Key insight: Maxwell's equations are among the most important equations in all of physics. They predicted electromagnetic waves 25 years before Hertz detected them experimentally.
Watch the capacitor charge, and pause or reset as needed. No charge crosses the gap, yet the growing E-field there makes the same magnetic field as a real current.
Current flows in the wire but cannot cross the capacitor gap. Maxwell realized that the changing electric field between the plates acts as a "displacement current," preserving current continuity and completing Ampere's law.
Maxwell's key insight: a changing electric field acts as a current (the displacement current). In a charging capacitor, no actual charge crosses the gap, but the growing E-field produces the same magnetic effect as a real current. This term completes Ampere's law and enables electromagnetic waves.
Key insight: Without the displacement current, Ampere's law would be inconsistent (the circulation of B would depend on which surface you use). Maxwell's correction restored consistency and predicted light.
Watch E and B take turns: each changing field creates the other, and the pair travels off together at the speed of light.
Faraday's law and the Ampere-Maxwell law form a feedback loop: a changing B field produces an E field, and a changing E field produces a B field. This self-sustaining oscillation propagates through space as an electromagnetic wave at speed c = 1/√(μ&sub0;ε&sub0;).
Faraday: changing B creates E. Ampere-Maxwell: changing E creates B. This mutual induction forms a self-sustaining oscillation that propagates through space at c = 1/√(μ₀ε₀). This is light.
Key insight: The speed of light emerges from the coupling constants of electricity (ε₀) and magnetism (μ₀). Maxwell realized that light is an electromagnetic wave — one of the greatest unifications in physics.
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