1
Maxwell’s Equations
1.1
Maxwell’s Equations
Maxwell’s equations describe all (classical) electromagnetic phenomena:
∇ ×
E
= −
∂
B
∂t
∇ ×
H
=
J
+
∂
D
∂t
∇ ·
D
=
ρ
∇ ·
B
=
0
(Maxwell’s equations)
(1.1.1)
The first is
Faraday’s law of induction
, the second is
Amp`
ere’s law
as amended by
Maxwell to include the displacement current
∂
D
/∂t
, the third and fourth are
Gauss’ laws
for the electric and magnetic fields.
The displacement current term
∂
D
/∂t
in Amp`
ere’s law is essential in predicting the
existence of propagating electromagnetic waves. Its role in establishing charge conser-
vation is discussed in Sec. 1.7.
Eqs. (1.1.1) are in SI units.
The quantities
E
and
H
are the electric and magnetic
field intensities
and are measured in units of [volt/m] and [ampere/m], respectively.
The quantities
D
and
B
are the electric and magnetic
ﬂux densities
and are in units of
[coulomb/m
2
] and [weber/m
2
], or [tesla].
D
is also called the
electric displacement
, and
B
, the
magnetic induction
.
The quantities
ρ
and
J
are the
volume charge density
and
electric current density
(charge ﬂux) of any
external
charges (that is, not including any induced polarization
charges and currents.) They are measured in units of [coulomb/m
3
] and [ampere/m
2
].
The right-hand side of the fourth equation is zero because there are no magnetic mono-
pole charges. Eqs. (1.3.17)–(1.3.19) display the induced polarization terms explicitly.
The charge and current densities
ρ,
J
may be thought of as the
sources
of the electro-
magnetic fields. For wave propagation problems, these densities are localized in space;
for example, they are restricted to ﬂow on an antenna. The generated electric and mag-
netic fields are
radiated
away from these sources and can propagate to large distances to
2
1.
Maxwell’s Equations
the receiving antennas. Away from the sources, that is, in source-free regions of space,
Maxwell’s equations take the simpler form:
∇ ×
E
= −
∂
B
∂t
∇ ×
H
=
∂
D
∂t
∇ ·
D
=
0
∇ ·
B
=
0
(source-free Maxwell’s equations)
(1.1.2)
The qualitative mechanism by which Maxwell’s equations give rise to propagating
electromagnetic fields is shown in the figure below.
For example, a time-varying current
J
on a linear antenna generates a circulating
and time-varying magnetic field
H
, which through Faraday’s law generates a circulating
electric field
E
, which through Amp`
ere’s law generates a magnetic field, and so on. The
cross-linked electric and magnetic fields propagate away from the current source.
A
more precise discussion of the fields radiated by a localized current distribution is given
in Chap. 15.
1.2
Lorentz Force
The force on a charge
q
moving with velocity
v
in the presence of an electric and mag-
netic field
E
,
B
is called the Lorentz force and is given by:
F
=
q(
E
+
v
×
B
)
(Lorentz force)
(1.2.1)
Newton’s equation of motion is (for non-relativistic speeds):
m
d
v
dt
=
F
=
q(
E
+
v
×
B
)
(1.2.2)
where
m
is the mass of the charge. The force
F
will increase the kinetic energy of the

#### You've reached the end of your free preview.

Want to read all 18 pages?

- Spring '14
- Energy, Electric charge, Permittivity, Dielectric, unit volume