Here E and B represent electric and magnetic fields, respectively. The charge density ρ describes the charge per volume distributed through space.3 The current density J describes the motion of charge density (in units of ρ times velocity). The constant ²0 is called the permittivity, and the constant µ0 is called the permeability. Taken together, these are known as Maxwell’s equations. After introducing a key revision of Ampere’s law, Maxwell realized that together these equations comprise a complete self-consistent theory of electromagnetic phenomena.
Moreover, the equations imply the existence of electromagnetic waves, which travel at the speed of light. Since the speed of light had been measured before Maxwell’s time, it was immediately apparent (as was already suspected) that light is a high-frequency manifestation of the same phenomena that govern the influence of currents and charges upon each other.
Previously, optics had been considered a topic quite separate from electricity and magnetism. Once the connection was made, it became clear that Maxwell’s equations form the theoretical foundations of optics, and this is where we begin our study of light.
This relationship is known as Coulomb’s law. The force is directed along the vector r−r 0 , which points from charge q 0 to q as seen in . 1.1. The length or magnitude of this vector is given by ¯ ¯r−r 0 ¯ ¯ (i.e. the distance between q 0 and q). The familiar inverse square law can be seen by noting that ¡ r−r 0 ¢±¯ ¯r−r 0 ¯ ¯ is a unit vector. We have written the force in terms of an electric field E(r), which is defined throughout space (regardless of whether a second charge q is actually present).
The permittivity ²0 amounts to a proportionality constant. Origi 1.1 The geometry of Coulomb’s law for a point charge. The total force from a collection of charges is found by summing the expression overall charges q 0 n associated with their specific locations r 0 n . If the charges are distributed continuously throughout space, having density ρ ¡ r 0 ¢ (units of charge per volume), the summation for finding the net electric field at r becomes an integral.
Gauss’ Law for Magnetic Fields
In order to ‘feel’ a magnetic force, a charge q must be moving at some velocity (call it v). The magnetic field arises itself from charges that are in motion. We consider the magnetic field to arise from the distribution of moving charges described by a current density J ¡ r 0 ¢ throughout space. The current density has units of charge times velocity per volume (or equivalently, current per cross-sectional area). The magnetic force law is analogous to Coulomb’s law.
Like Coulomb’s law, the Biot Savart law is incomplete since it also implies an instantaneous response of the magnetic field to a reconfiguration of the currents. The generalized version of the Biot-Savart law, another of Jefimenko’s equations, incorporates the fact that electromagnetic news travels at the speed of light. Ironically, Gauss’ law for magnetic fields and Maxwell’s version of Ampere’s law, derived from the Biot-Savart law, hold perfectly whether the currents are steady or vary in time.
The Jefimenko equations, analogs of Coulomb and Biot-Savart, also embody Faraday’s law, the only of Maxwell’s equations that cannot be derived from the usual forms of Coulomb’s law and the Biot-Savart law. See D. J. Griffiths, Introduction to Electrodynamics, 3rd ed., Sect. 10.2.2 (New Jersey: Prentice-Hall, 1999).