:<math{h>\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> ▼{{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]]. 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]].
Fundamental physical aspects of classical electrodynamics are presented in many texts, such as those by [[Richard Feynman|Feynman]], [[Robert B. Leighton|Leighton]] and [[Matthew Sands|Sands]],<ref>Feynman, R. P., R .B. Leighton, and M. Sands, 1965, ''[[The Feynman Lectures on Physics]], Vol. II: the Electromagnetic Field'', Addison-Wesley, Reading, Massachusetts</ref> [[David J. Griffiths|Griffiths]],<ref>{{cite book|last1=Griffiths|first1=David J.|title=Introduction to Electrodynamics|date=2013|publisher=Pearson|location=Boston, Mas.|isbn=978-0321856562|edition=4th}}</ref> [[Wolfgang K. H. Panofsky|Panofsky]] and Phillips,<ref>Panofsky, W. K., and M. Phillips, 1969, ''Classical Electricity and Magnetism'', 2nd edition, Addison-Wesley, Reading, Massachusetts</ref> and [[John David Jackson (physicist)|Jackson]].<ref name="Jack">{{Cite book|last=Jackson|first=John D.|title=Classical Electrodynamics|publisher=Wiley|location=New York|year=1998|edition=3rd|isbn=978-0-471-30932-1|title-link=Classical Electrodynamics (book)}}</ref>
== 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]]. For a detailed historical account, consult Pauli,<ref>Pauli, W., 1958, ''Theory of Relativity'', Pergamon, London</ref> Whittaker,<ref>Whittaker, E. T., 1960, ''History of the Theories of the Aether and Electricity'', Harper Torchbooks, New York.</ref> Pais,<ref>Pais, A., 1983, ''[[Subtle is the Lord: The Science and the Life of Albert Einstein]]'', Oxford University Press, Oxford</ref> and 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 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 {{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.
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