The four equations unified, displacement current, and symmetry
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.
Click any equation card to see an animated visual interpretation.
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.
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.
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.
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.
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.
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;).
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.