We develop Maxwell's equations from their experimental foundations in differential and integral form for both static and dynamic applications. We consider first the microscopic and then the macroscopic forms of the equations. Maxwell's equations are numbered in what follows.
Stationary point charges exert a mutual central force on one another. The force exerted on charge under the influence of in Figure B.1 is governed by the following empirical law:
and is attributed to the electric field established by:
In MKS units, and. The electric (Coulomb) force is a central force with a dependence on radial distance. The contributions from n point charges add linearly and exert the total force on the ith particle:
where we note that a point charge exerts no self force. For a continuous charge distribution, the electric field becomes
where the primed and unprimed coordinates denote the source and observation points, respectively, and where is the volume charge density, with MKS units of Coulombs per cubic meter. Substituting recovers the electric field due to n discrete, static point charges.
Evaluating Coulomb's law for any but the simplest charge distributions can be cumbersome and requires specification of the charge distribution a priori. An alternative and often more useful relationship between electric fields and charge distributions is provided by Gauss’ law. Gauss’ law will become part of a system of differential equations that permit the evaluation of the electric field, even before the charge distribution is known under some circumstances.
Consider the electric flux through the closed surface S surrounding the volume V containing the point charge q, shown in Figure B.2. According to Coulomb's law, the flux through the differential surface area element da is
where r is the distance from the point charge to the surface area element, is the unit normal of da and is the angle between and E.