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This document provides an overview of Maxwell's equations, which describe the electromagnetic field. [1] Maxwell's equations relate the divergence and curl of five vector fields - the electric field E, magnetic field B, electric current density j, electric displacement D, and magnetic field intensity H. [2] The equations are supplemented by two scalar relations involving the electric charge density ρ and the divergence of B. [3] Maxwell's equations show that electromagnetic waves can propagate through space and time at the speed of light c.

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89 views1 page

First Page PDF

This document provides an overview of Maxwell's equations, which describe the electromagnetic field. [1] Maxwell's equations relate the divergence and curl of five vector fields - the electric field E, magnetic field B, electric current density j, electric displacement D, and magnetic field intensity H. [2] The equations are supplemented by two scalar relations involving the electric charge density ρ and the divergence of B. [3] Maxwell's equations show that electromagnetic waves can propagate through space and time at the speed of light c.

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CHAPTER 1

BASIC P R O P E R T I E S OF T H E
ELECTROMAGNETIC FIELD

1.1 THE ELECTROMAGNETIC FIELD

1.1.1 M a x w e l l ' s equations


T H E state of excitation which is established in space by the presence of electric
charges is said to constitute an electromagnetic field. I t is represented by t w o vectors,
Ε and B , called the electric vector and the magnetic induction respectively.*
T o describe the effect of the field on material objects, it is necessary to introduce
a second set of vectors, viz. the electric current density j , the electric displacement D,
and the magnetic vector H.
The space and time derivatives of the five vectors are related by Maxwell's equations,
which hold at every point in whose neighbourhood the physical properties of the
medium are continuous:!

curl i f - i l > = — /J , (1)


c c

1 .
curl Ε + - Β = 0, (2)
c

the dot denoting differentiation with respect to time.


They are supplemented by two scalar relations :

div D = 4πρ, (3)

div Β = 0. (4)

Eq. ( 3 ) may be regarded as a defining equation for the electric charge density ρ and (4)
may be said to imply that no free magnetic poles exist.

* In elementary considerations Ε and Η are, for historical reasons, usually regarded as the basic
field vectors, and D and Β as describing the influence of matter. In general theory, however, the
present interpretation is compulsory for reasons connected with the electrodynamics of moving media.
The four M A X W E L L equations (l)-(4) can be divided into two sets of equations, one consisting
of two homogeneous equations (right-hand side zero), containing Ε and B , the other of two non-
homogeneous equations (right-hand side different from zero), containing D and H. If a co-
ordinate transformation of space and time (relativistio LORENTZ transformation) is carried out,
the equations of each group transform together, the equations remaining unaltered in form if//c
and p are transformed as a four-vector, and each of the pairs Ε, Β and D, H as a six-vector
(antisymmetric tensor of the second order). Since the nonhomogeneous set contains charges and
currents (which represent influence of matter), one has to attribute the corresponding pair ( D , H)
to the influence of matter. It is, however, customary to refer to Η and not to Β as the magnetic
field vector', we shall conform to this terminology when there is no risk of confusion.
t The so-called Gaussian system of units is used here, i.e. the electrical quantities (JE, D , / and p)
are measured in electrostatic units, and the magnetic quantities (H and B ) in electromagnetic
units. The constant c in (1) and (2) relates the unit of charge in10
the two systems; it is the velocity
of light in the vacuum and is approximately equal to 3 χ 10 cm/sec. (A more accurate value is
given in § 1.2.)
1

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