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The fundamental equations of classical macroscopic electrodynamics that describe electromagnetic phenomena in any medium. The equations were formulated by J. Maxwell in the 1860’s on the basis of a generalization of the empirical laws of electric and magnetic phenomena. By using these laws as a basis and developing M.

Faraday’s productive idea that the interactions between electrically charged bodies take place through an electromagnetic field, Maxwell created the theory of electromagnetic processes, which is expressed mathematically by Maxwell’s equations. The present form of the equations was given by the German physicist H. Hertz and the British physicist O.

Maxwell’s equations relate the quantities that characterize an electromagnetic field to its sources, that is, to the spatial distribution of electric charges and currents. In a vacuum, the electromagnetic field is characterized by two vector quantities that are dependent on spatial coordinates and on time—the electric field intensity E and magnetic induction B. These quantities determine the forces that act because of the field on the charges and currents whose distribution in space is given by the charge density ρ (the charge per unit volume) and the current density j (the charge passing in unit time through unit area perpendicular to the direction of motion of the charges). In addition to the vectors E and B, auxiliary vector quantities that are dependent on the state and properties of the medium—the electric displacement D and the magnetic field intensity H—are introduced to describe electromagnetic processes in a material medium (matter).

Maxwell’s equations make it possible to determine the fundamental characteristics of a field ( E, B, D, and H) at each point in space at any moment if the field sources j and ρ are known as functions of the coordinates and of time. The equations can be written in integral or differential form [below they are given in the absolute (Gaussian) system of units]. Maxwell’s equations in integral form determine on the basis of given charges and currents not the field vectors E, B, D and H themselves at different points in space but certain integral quantities that depend on the distribution of these field characteristics: the line integral (circulation) of the vectors E and H around any closed curve and the surface integral (flux) of the vectors D and B through any closed surface. Maxwell’s first equation is a generalization for variable fields of the empirical law of Ampere which deals with the excitation of a magnetic field by an electric current. Maxwell advanced the hypothesis that magnetic fields are generated not only by currents flowing in conductors but also by varying electric fields in dielectrics or a vacuum.

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A quantity proportional to the rate of change of the electric field with time was called displacement current by Maxwell. A displacement current excites a magnetic field by the same law as does a conduction current (this was confirmed later experimentally). The total current, which is equal to the sum of the conduction current and the displacement current, is always closed.

Maxwell’s first equation has the form: that is, the line integral of the magnetic field vector around a closed curve L (the sum of the scalar products of the vector H at a given point in the circuit and an infinitesimal segment dl of the curve) is given by the total current through any surface S bounded by the curve. Here j n is the projection of the conduction current density j on the normal to the infinitesimal area ds, which is a part of the surface S; (1/4 π)(∂ D n/∂ t) is the projection of the displacement current density on the same normal; and c = 3 × 10 10 cm/sec is a constant equal to the rate of propagation of electromagnetic interactions in a vacuum. Maxwell’s second equation is a mathematical formulation of Faraday’s law of elecromagnetic induction and is written in the form that is, the line integral of the electric field vector along a closed curve L (the electromotive force of induction) is given by the rate of change of the magnetic induction through the surface S bounded by the curve.

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