# 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. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell proposed that light (including radiant heat, and other radiations if any) is an electromagnetic phenomenon, based upon deriving of the speed of light from his equations.

• ... 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.
• ... 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.
• Whenever I teach a course on electromagnetism, one of the first ... exam questions I will ask students is, "Why is the magnetic field called $H$  in the textbook?" ... The reason it is called $H$  is, of course, Maxwell did not use the vector notation. He writes out all the components. So the electric field he starts with $E$  ... And so, the electric field is $E,F,G,$  and the magnetic field is, obviously, $H,I,J.$  ... So all six components are written out. And so, it's very misleading for a textbook nowadays to say there are four Maxwell's equations. There were actually, if you look at Maxwell's papers, something like twenty-four equations. And the whole thing looks incredibly complicated.