Could not include differential geometry - contents
In the context of electromagnetism, Maxwell’s equations are the equations of motion for the electromagnetic field strength electric current and magnetic current.
Three dimensional formulation
is here the (vector of) strength of electric field and the strength of magnetic field; is the charge and the density of the electrical current; , , are constants (electrical permeability, speed of the light, and magnetic permeability; all of/in vacuum: ).
Integral formulation in vacuum, in SI units
Gauss’ law for electric fields
where is a closed surface which is a boundary of a 3d domain (physicists say “volume”) and the charge in the domain ; denotes the scalar (dot) product. Surface element is , i.e. it is the scalar surface measure times the unit vector of normal outwards.
No magnetic monopoles (Gauss’ law for magnetic fields)
where is any closed surface.
Faraday’s law of induction
The line element is the differential (or 1-d measure on the boundary) of the length times the unit vector in counter-circle direction (or parametrize the curve with being a vector in 3d space, express magnetic field in the same parameter and calculate the integral as a function of parameter: is a scalar (“dot”) product).
Ampère-Maxwell law (or generalized Ampère’s law; Maxwell added the second term involving derivative of the flux of electric field to the Ampère’s law which described the magnetic field due electric current).
where is a surface and its boundary; is the total current through (integral of the component of normal to the surface).
Here we put units with . By we denote the density of the charge.
no magnetic charges (magnetic Gauss law):
generalized Ampère’s law
Equations in terms of Faraday tensor
This is adapted from electromagnetic field – Maxwell’s equations.
In modern language, the insight of (Maxwell, 1865) is that locally, when physical spacetime is well approximated by a patch of its tangent space, i.e. by a patch of 4-dimensional Minkowski space , the electric field and magnetic field combine into a differential 2-form
in and the electric charge density and current density combine to a differential 3-form
in such that the following two equations of differential forms are satisfied
where is the de Rham differential operator and the Hodge star operator. If we decompose into its components as before as
then in terms of these components the field equations – called Maxwell’s equations – read as follows.
gives the magnetic Gauss law and Faraday’s law
gives Gauss’s law and Ampère-Maxwell law
Maxwell's equations originate in
Discussion in terms of differential forms is for instance in
- Theodore Frankel, Maxwell’s equations, The American Mathematical Monthly, Vol 81, No 4 (1974) (pdf, JSTOR)
- section 3.5 in Theodore Frankel, The Geometry of Physics - An Introduction
- Gregory L. Naber, Topology, geometry and gauge fields, Appl. Math. Sciences vol. 141, Springer 2000