Classical electromagnetism or classical electrodynamics
is a branch of theoretical physics that studies
the interactions between electric charges
and currents using an extension of the classical
Newtonian model. The theory provides an excellent
description of electromagnetic phenomena whenever
the relevant length scales and field strengths
are large enough that quantum mechanical effects
are negligible. For small distances and low
field strengths, such interactions are better
described by quantum electrodynamics.
Fundamental physical aspects of classical
electrodynamics are presented in many texts,
such as those by Feynman, Leighton and Sands,
Griffiths, Panofsky and Phillips, and Jackson.
== History ==
The physical phenomena that electromagnetism
describes have been studied as separate fields
since antiquity. For example, there were many
advances in the field of optics centuries
before light was understood to be an electromagnetic
wave. However, the theory of electromagnetism,
as it is currently understood, grew out of
Michael Faraday's experiments suggesting an
electromagnetic field and James Clerk Maxwell's
use of differential equations to describe
it in his A Treatise on Electricity and Magnetism
(1873). For a detailed historical account,
consult Pauli, Whittaker, Pais, and Hunt.
== Lorentz force ==
The electromagnetic field exerts the following
force (often called the Lorentz force) on
charged particles:
F
=
q
E
+
q
v
×
B
{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf
{v} \times \mathbf {B} }
where all boldfaced quantities are vectors:
F is the force that a particle with charge
q experiences, E is the electric field at
the location of the particle, v is the velocity
of the particle, B is the magnetic field at
the location of the particle.
The above equation illustrates that the Lorentz
force is the sum of two vectors. One is the
cross product of the velocity and magnetic
field vectors. Based on the properties of
the cross product, this produces a vector
that is perpendicular to both the velocity
and magnetic field vectors. The other vector
is in the same direction as the electric field.
The sum of these two vectors is the Lorentz
force.
Therefore, in the absence of a magnetic field,
the force is in the direction of the electric
field, and the magnitude of the force is dependent
on the value of the charge and the intensity
of the electric field. In the absence of an
electric field, the force is perpendicular
to the velocity of the particle and the direction
of the magnetic field. If both electric and
magnetic fields are present, the Lorentz force
is the sum of both of these vectors.
Although the equation appears to suggest that
the Electric and Magnetic fields are independent,
the equation can be rewritten in term of four-current
(instead of charge) and a single tensor that
represents the combined Electromagnetic field
(
F
μ
ν
{\displaystyle F^{\mu \nu }}
)
f
α
=
F
α
β
J
β
.
{\displaystyle f_{\alpha }=F_{\alpha \beta
}J^{\beta }.\!}
== The electric field E ==
The electric field E is defined such that,
on a stationary charge:
F
=
q
0
E
{\displaystyle \mathbf {F} =q_{0}\mathbf {E}
}
where q0 is what is known as a test charge.
The size of the charge doesn't really matter,
as long as it is small enough not to influence
the electric field by its mere presence. What
is plain from this definition, though, is
that the unit of E is N/C (newtons per coulomb).
This unit is equal to V/m (volts per meter);
see below.
In electrostatics, where charges are not moving,
around a distribution of point charges, the
forces determined from Coulomb's law may be
summed. The result after dividing by q0 is:
E
(
r
)
=
1
4
π
ε
0
∑
i
=
1
n
q
i
(
r
−
r
i
)
|
r
−
r
i
|
3
{\displaystyle \mathbf {E(r)} ={\frac {1}{4\pi
\varepsilon _{0}}}\sum _{i=1}^{n}{\frac {q_{i}\left(\mathbf
{r} -\mathbf {r} _{i}\right)}{\left|\mathbf
{r} -\mathbf {r} _{i}\right|^{3}}}}
where n is the number of charges, qi is the
amount of charge associated with the ith charge,
ri is the position of the ith charge, r is
the position where the electric field is being
determined, and ε0 is the electric constant.
If the field is instead produced by a continuous
distribution of charge, the summation becomes
an integral:
E
(
r
)
=
1
4
π
ε
0
∫
ρ
(
r
′
)
(
r
−
r
′
)
|
r
−
r
′
|
3
d
3
r
′
{\displaystyle \mathbf {E(r)} ={\frac {1}{4\pi
\varepsilon _{0}}}\int {\frac {\rho (\mathbf
{r'} )\left(\mathbf {r} -\mathbf {r'} \right)}{\left|\mathbf
{r} -\mathbf {r'} \right|^{3}}}\mathrm {d^{3}}
\mathbf {r'} }
where
ρ
(
r
′
)
{\displaystyle \rho (\mathbf {r'} )}
is the 
charge density and
r
−
r
′
{\displaystyle \mathbf {r} -\mathbf {r'} }
is the vector that points from the volume
element
d
3
r
′
{\displaystyle \mathrm {d^{3}} \mathbf {r'}
}
to the point in space where E is being determined.
Both of the above equations are cumbersome,
especially if one wants to determine E as
a function of position. A scalar function
called the electric potential can help. Electric
potential, also called voltage (the units
for which are the volt), is defined by the
line integral
φ
(
r
)
=
−
∫
C
E
⋅
d
l
{\displaystyle \varphi \mathbf {(r)} =-\int
_{C}\mathbf {E} \cdot \mathrm {d} \mathbf
{l} }
where φ(r) is the electric potential, and
C is the path over which the integral is being
taken.
Unfortunately, this definition has a caveat.
From Maxwell's equations, it is clear that
∇ × E is not always zero, and hence the
scalar potential alone is insufficient to
define the electric field exactly. As a result,
one must add a correction factor, which is
generally done by subtracting the time derivative
of the A vector potential described below.
Whenever the charges are quasistatic, however,
this condition will be essentially met.
From the definition of charge, one can easily
show that the electric potential of a point
charge as a function of position is:
φ
(
r
)
=
1
4
π
ε
0
∑
i
=
1
n
q
i
|
r
−
r
i
|
{\displaystyle \varphi \mathbf {(r)} ={\frac
{1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac
{q_{i}}{\left|\mathbf {r} -\mathbf {r} _{i}\right|}}}
where q is the point charge's charge, r is
the position at which the potential is being
determined, and ri is the position of each
point charge. The potential for a continuous
distribution of charge is:
φ
(
r
)
=
1
4
π
ε
0
∫
ρ
(
r
′
)
|
r
−
r
′
|
d
3
r
′
{\displaystyle \varphi \mathbf {(r)} ={\frac
{1}{4\pi \varepsilon _{0}}}\int {\frac {\rho
(\mathbf {r'} )}{|\mathbf {r} -\mathbf {r'}
|}}\,\mathrm {d^{3}} \mathbf {r'} }
where
ρ
(
r
′
)
{\displaystyle \rho (\mathbf {r'} )}
is the charge density, and
r
−
r
′
{\displaystyle \mathbf {r} -\mathbf {r'} }
is the distance from the volume element
d
3
r
′
{\displaystyle \mathrm {d^{3}} \mathbf {r'}
}
to point in space where φ is being determined.
The scalar φ will add to other potentials
as a scalar. This makes it relatively easy
to break complex problems down in to simple
parts and add their potentials. Taking the
definition of φ backwards, we see that the
electric field is just the negative gradient
(the del operator) of the potential. Or:
E
(
r
)
=
−
∇
φ
(
r
)
.
{\displaystyle \mathbf {E(r)} =-\nabla \varphi
\mathbf {(r)} .}
From this formula it is clear that E can be
expressed in V/m (volts per meter).
== Electromagnetic waves ==
A changing electromagnetic field propagates
away from its origin in the form of a wave.
These waves travel in vacuum at the speed
of light and exist in a wide spectrum of wavelengths.
Examples of the dynamic fields of electromagnetic
radiation (in order of increasing frequency):
radio waves, microwaves, light (infrared,
visible light and ultraviolet), x-rays and
gamma rays. In the field of particle physics
this electromagnetic radiation is the manifestation
of the electromagnetic interaction between
charged particles.
== General field equations ==
As simple and satisfying as Coulomb's equation
may be, it is not entirely correct in the
context of classical electromagnetism. Problems
arise because changes in charge distributions
require a non-zero amount of time to be "felt"
elsewhere (required by special relativity).
For the fields of general charge distributions,
the retarded potentials can be computed and
differentiated accordingly to yield Jefimenko's
equations.
Retarded potentials can also be derived for
point charges, and the equations are known
as the Liénard–Wiechert potentials. The
scalar potential is:
φ
=
1
4
π
ε
0
q
|
r
−
r
q
(
t
r
e
t
)
|
−
v
q
(
t
r
e
t
)
c
⋅
(
r
−
r
q
(
t
r
e
t
)
)
{\displaystyle \varphi ={\frac {1}{4\pi \varepsilon
_{0}}}{\frac {q}{\left|\mathbf {r} -\mathbf
{r} _{q}(t_{ret})\right|-{\frac {\mathbf {v}
_{q}(t_{ret})}{c}}\cdot (\mathbf {r} -\mathbf
{r} _{q}(t_{ret}))}}}
where q is the point charge's charge and r
is the position. rq and vq are the position
and velocity of the charge, respectively,
as a function of retarded time. The vector
potential is similar:
A
=
μ
0
4
π
q
v
q
(
t
r
e
t
)
|
r
−
r
q
(
t
r
e
t
)
|
−
v
q
(
t
r
e
t
)
c
⋅
(
r
−
r
q
(
t
r
e
t
)
)
.
{\displaystyle \mathbf {A} ={\frac {\mu _{0}}{4\pi
}}{\frac {q\mathbf {v} _{q}(t_{ret})}{\left|\mathbf
{r} -\mathbf {r} _{q}(t_{ret})\right|-{\frac
{\mathbf {v} _{q}(t_{ret})}{c}}\cdot (\mathbf
{r} -\mathbf {r} _{q}(t_{ret}))}}.}
These can then be differentiated accordingly
to obtain the complete field equations for
a moving point particle.
== Models ==
Branches of classical electromagnetism such
as optics, electrical and electronic engineering
consist of a collection of relevant mathematical
models of different degrees of simplification
and idealization to enhance the understanding
of specific electrodynamics phenomena, cf.
An electrodynamics phenomenon is determined
by the particular fields, specific densities
of electric charges and currents, and the
particular transmission medium. Since there
are infinitely many of them, in modeling there
is a need for some typical, representative
(a) electrical charges and currents, e.g.
moving pointlike charges and electric and
magnetic dipoles, electric currents in a conductor
etc.;
(b) electromagnetic fields, e.g. voltages,
the Liénard–Wiechert potentials, the monochromatic
plane waves, optical rays; radio waves, microwaves,
infrared radiation, visible light, ultraviolet
radiation, X-rays, gamma rays etc.;
(c) transmission media, e.g. electronic components,
antennas, electromagnetic waveguides, flat
mirrors, mirrors with curved surfaces convex
lenses, concave lenses; resistors, inductors,
capacitors, switches; wires, electric and
optical cables, transmission lines, integrated
circuits etc.;all of which have only few variable
characteristics.
== See also ==
Electromagnetism
Maxwell's equations
Weber electrodynamics
Wheeler–Feynman absorber theory
Leontovich boundary condition
