# Maxwell's equations

set of partial differential equations that describe how electric and magnetic fields are generated and altered by each other and by charges and currents

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.

## Quotes

• But the mathematicians of the nineteenth century failed miserably to grasp the equally great opportunity offered to them in 1865 by Maxwell. If they had taken Maxwell's equations to heart as Euler took Newton's, they would have discovered, among other things, Einstein's theory of special relativity, the theory of topological groups and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis. A great part of twentieth century physics and mathematics could have been created in the nineteenth century, simply by exploring to the end the mathematical concepts to which Maxwell's equations naturally lead.
• We have a wave which leaves the material source and goes outward at the velocity c, which is the speed of light. ... From a historical point of view, it wasn’t known that the coefficient c in Maxwell’s equations was also the speed of light propagation. There was just a constant in the equations. We have called it c from the beginning, because we knew what it would turn out to be. We didn’t think it would be sensible to make you learn the formulas with a different constant and then go back to substitute c wherever it belonged. ... just by experiments with charges and currents we find a number c2 which turns out to be the square of the velocity of propagation of electromagnetic influences. From static measurements—by measuring the forces between two unit charges and between two unit currents—we find that c = 3.00 × 108 meters/sec. When Maxwell first made this calculation with his equations, he said that bundles of electric and magnetic fields should be propagated at this speed. He also remarked on the mysterious coincidence that this was the same as the speed of light. “We can scarcely avoid the inference,” said Maxwell, “that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” Maxwell had made one of the great unifications of physics. Before his time, there was light, and there was electricity and magnetism. The latter two had been unified by the experimental work of Faraday, Oersted, and Ampère. Then, all of a sudden, light was no longer “something else,” but was only electricity and magnetism in this new form—little pieces of electric and magnetic fields which propagate through space on their own.
• ... the fullest statement Maxwell gave of his theory, his 1873 Treatise on Electricity and Magnetism, does not contain the four famous "Maxwell's equations," nor does it even hint at how electromagnetic waves might be produced or detected. These and many other aspects of the theory were quite thoroughly hidden in the version of it given by Maxwell himself; in the words of Oliver Heaviside, they were "latent" but hardly "patent." ... Maxwell was only forty-eight when he died of cancer in November 1879. ... the task of digging out the "latent" aspects of his theory and of exploring its wider implications was thus left to a group of younger physicists, most of them British.
• ... the orginal field equations explicitly contain the magnetic vector potential, ${\overrightarrow {A}}$  ... In Maxwell's original formulaton, Faraday's ${\overrightarrow {A}}$  field was central and had physical meaning. The magnetic vector potential was not arbitrary, as defined by boundary conditions and choice of gauge as we will discuss; they were said to be gauge invariant. The original equations are thus often called the Faraday-Maxwell theory.
• ... applications in gauge field theories and the physics of condensed matter. The starting point here is now quite well known: expressing the Maxwell equations for an electromagnetic field over Lorentz space as the Euler-Lagrange equations for a Lagrangian defined on the connections on a $U(1)$  bundle, where the electromagnetic potential becomes the connection and the field tensor its curvature. The freedom of choice of "gauge" for the potential is a fundamental fact which stems, in the geometrical picture, from the lack of a preferred trivialisation of the bundle.