Maxwell's equations
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Electricity, magnetism, radio, light, X-rays, and the warmth of sunlight are all the same thing — an electromagnetic field — and four PDEs describe every bit of it. Maxwell's equations are the rule book for how electric and magnetic fields are made by charges and currents, how each one creates and feeds the other, and how, set free, they ripple away as waves. They are one of the supreme triumphs of nineteenth-century physics and a perfect showcase of PDEs running the modern world.
Let E be the electric field and B the magnetic field. In a vacuum, the four equations are: div E = rho / epsilon0 (Gauss's law — charges are the sources of E); div B = 0 (no magnetic monopoles — B has no sources); curl E = -B_t (Faraday's law — a changing magnetic field drives an electric field); and curl B = mu0 J + mu0 epsilon0 E_t (Ampere-Maxwell law — currents and a changing electric field drive a magnetic field). The magic is in the cross-coupling: take the curl of Faraday's law and feed in Ampere's, and in empty space you derive the wave equation E_tt = c^2 Laplacian E, with c = 1/sqrt(mu0 epsilon0) coming out as the speed of light. That single algebraic step told Maxwell that light is an electromagnetic wave.
These equations underlie all of electrical engineering, optics, antennas, fibre-optic communication, and your phone's radio. In the static case the time-derivatives vanish and they reduce to familiar elliptic PDEs: the electrostatic potential satisfies Poisson's equation Laplacian phi = -rho / epsilon0. So Maxwell's equations contain the wave equation, Poisson's equation, and Laplace's equation as special cases — a beautiful demonstration of how one physical law splits into the canonical PDE types depending on whether you ask about waves, sources, or steady fields.
In a charge-free, current-free region, both E and B satisfy the wave equation, e.g. E_tt = c^2 Laplacian E. A plane-wave solution E = E0 cos(k dot x - omega t) needs omega = c |k|: it travels at speed c. That this 'c' equals the measured speed of light is how electromagnetism and optics became one subject.
Four PDEs that contain the wave, Poisson, and Laplace equations as special cases.
The compact 'four equations' form is the macroscopic vacuum version; inside materials you carry extra fields D and H and constitutive relations, and the truly fundamental form is relativistic and gauge-theoretic. The vacuum quartet is exact but not the whole story.