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{{redirect|Classical electrodynamics|the textbook by J. D. Jackson|Classical Electrodynamics (book)}}
{{short description|Branch of theoretical physics}}
{{electromagnetism|cTopic=Electrodynamics}}
'''Classical electromagnetism''' or '''classical electrodynamics''' is a branch of [[theoretical physics]] that studies the interactions between [[electric charge]]s and [[electrical current|currents]] using an extension of the [[classical Newtonian model]]. It is, therefore, a [[classical field theory]]. The theory provides a description of electromagnetic phenomena whenever the relevant [[length scale]]s 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]] which is a [[quantum field theory]].

Fundamental physical aspects of classical electrodynamics are presented in many textbooks. For the undergraduate level, textbooks like ''[[The Feynman Lectures on Physics]]'', [[Electricity and Magnetism (book)|''Electricity and Magnetism'']], and ''[[Introduction to Electrodynamics]]'' are considered as classic references and for the graduate level, textbooks like ''Classical Electricity and Magnetism'',<ref>{{cite book | last1=Panofsky | first1=W. K. H. | author1-link=Pief Panofsky | last2=Phillips | first2=M. | author2-link=Melba Phillips | title=Classical Electricity and Magnetism | publisher=[[Dover Publications|Dover]] |date=2005 | isbn=9780486439242 | url=https://store.doverpublications.com/0486439240.html}}</ref> [[Classical Electrodynamics (book)|''Classical Electrodynamics'']], and ''[[Course of Theoretical Physics]]'' are considered as classic references.

== History ==
{{Main|History of electromagnetism}}

The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of [[History of optics|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 the existence of an [[electromagnetic field]] and [[James Clerk Maxwell]]'s use of [[differential equation]]s to describe it in his ''[[A Treatise on Electricity and Magnetism]]'' (1873). The development of electromagnetism in Europe included the development of methods to measure [[voltage]], [[Electric current|current]], [[capacitance]], and [[Electrical resistance and conductance|resistance]]. Detailed historical accounts are given by [[Wolfgang Pauli]],<ref>Pauli, W., 1958, ''Theory of Relativity'', Pergamon, London</ref> [[E. T. Whittaker]],<ref>Whittaker, E. T., 1960, ''History of the Theories of the Aether and Electricity'', Harper Torchbooks, New York.</ref> [[Abraham Pais]],<ref>Pais, A., 1983, ''[[Subtle is the Lord: The Science and the Life of Albert Einstein]]'', Oxford University Press, Oxford</ref> and Bruce J. Hunt.<ref>Bruce J. Hunt (1991) [[The Maxwellians]]</ref>

== Lorentz force ==
{{Main|Lorentz force}}

The electromagnetic field exerts the following force (often called the Lorentz force) on [[Electric charge|charged]] particles:

:<math>
\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B}
</math>

where all boldfaced quantities are [[Vector (geometric)|vectors]]: {{math|'''F'''}} is the force that a particle with charge ''q'' experiences, {{math|'''E'''}} is the [[electric field]] at the location of the particle, {{math|'''v'''}} is the velocity of the particle, {{math|'''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.

Although the equation appears to suggest that the electric and magnetic fields are independent, the equation [[Covariant formulation of classical electromagnetism#Lorentz force|can be rewritten]] in term of [[four-current]] (instead of charge) and a single [[electromagnetic tensor]] that represents the combined field (<math>F^{\mu \nu}</math>):
:<math>f_{\alpha} = F_{\alpha\beta}J^{\beta} .\!</math>

==Electric field==
{{Main|Electric field}}

The electric field '''E''' is defined such that, on a stationary charge:

:<math>
\mathbf{F} = q_0 \mathbf{E}
</math>

where ''q''<sub>0</sub> is what is known as a test charge and {{math|'''F'''}} is the [[Electrostatic force|force]] on that charge. The size of the charge does not 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 {{math|'''E'''}} is N/C ([[newton (unit)|newtons]] per [[coulomb]]). This unit is equal to V/m ([[volt]]s 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 ''q''<sub>0</sub> is:

:<math>\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}</math>

where ''n'' is the number of charges, ''q<sub>i</sub>'' is the amount of charge associated with the ''i''th charge, '''r'''<sub>''i''</sub> is the position of the ''i''th charge, '''r''' is the position where the electric field is being determined, and ''ε''<sub>0</sub> is the [[electric constant]].

If the field is instead produced by a continuous distribution of charge, the summation becomes an integral:

:<math>\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'}</math>

where <math>\rho(\mathbf{r'})</math> is the [[charge density]] and <math>\mathbf{r}-\mathbf{r'}</math> is the vector that points from the volume element <math>\mathrm{d^3}\mathbf{r'}</math> 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]]

:<math>
\varphi \mathbf{(r)} = - \int_C \mathbf{E} \cdot \mathrm{d}\mathbf{l}
</math>

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 {{nowrap|∇ × '''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:

:<math>
\varphi \mathbf{(r)} = \frac{1}{4 \pi \varepsilon_0 }
\sum_{i=1}^{n} \frac{q_i} {\left| \mathbf{r} - \mathbf{r}_i \right|}
</math>

where ''q'' is the point charge's charge, '''r''' is the position at which the potential is being determined, and '''r'''<sub>''i''</sub> is the position of each point charge. The potential for a continuous distribution of charge is:

:<math>
\varphi \mathbf{(r)} = \frac{1}{4 \pi \varepsilon_0}
\int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}\, \mathrm{d^3}\mathbf{r'}
</math>

where <math>\rho(\mathbf{r'})</math> is the charge density, and <math>\mathbf{r}-\mathbf{r'}</math> is the distance from the volume element <math>\mathrm{d^3}\mathbf{r'}</math> 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 into 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:

:<math> \mathbf{E(r)} = -\nabla \varphi \mathbf{(r)} .</math>

From this formula it is clear that '''E''' can be expressed in V/m (volts per meter).

== Electromagnetic waves ==
{{Main|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 [[electromagnetic spectrum|spectrum]] of [[wavelength]]s. Examples of the dynamic fields of [[electromagnetic radiation]] (in order of increasing frequency): [[radio wave]]s, [[microwave]]s, [[light]] ([[infrared]], [[visible light]] and [[ultraviolet]]), [[x-ray]]s 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 ==
{{Main|Jefimenko's equations|Liénard–Wiechert potential}}

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:

:<math>
\varphi = \frac{1}{4 \pi \varepsilon_0} \frac{q}{\left| \mathbf{r} - \mathbf{r}_q(t_{\rm ret}) \right|-\frac{\mathbf{v}_q(t_{\rm ret})}{c} \cdot (\mathbf{r} - \mathbf{r}_q(t_{\rm ret}))}
</math>

where ''q'' is the point charge's charge and '''r''' is the position. '''r'''<sub>''q''</sub> and '''v'''<sub>''q''</sub> are the position and velocity of the charge, respectively, as a function of [[retarded time]]. The [[vector potential]] is similar:

:<math>
\mathbf{A} = \frac{\mu_0}{4 \pi} \frac{q\mathbf{v}_q(t_{\rm ret})}{\left| \mathbf{r} - \mathbf{r}_q(t_{\rm ret}) \right|-\frac{\mathbf{v}_q(t_{\rm ret})}{c} \cdot (\mathbf{r} - \mathbf{r}_q(t_{\rm ret}))}.
</math>

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 model]]s of different degrees of simplification and idealization to enhance the understanding of specific electrodynamics phenomena.<ref>[[Rudolf Peierls|Peierls]], Rudolf. Model-making in physics, Contemporary Physics, Volume 21 (1), January 1980, 3-17.</ref> 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 ==
* [[List of textbooks in electromagnetism]]
* [[Leontovich boundary condition]]
* [[Weber electrodynamics]]
* [[Wheeler–Feynman absorber theory]]

== References ==
{{Reflist}}
<references responsive="0" />

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