en>Jonesey95: Fixing "Pages with citations using unnamed parameters" error.

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Fixing "Pages with citations using unnamed parameters" error.

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{{electromagnetism}}
{{Multiple issues|{{lead rewrite|date=March 2013|reason=Far too much technical information in the lead}}{{lead too long|date=March 2013}}{{technical|date=March 2013}}}}

In [[electromagnetism]], a branch of fundamental [[physics]], the '''matrix representations of the [[Maxwell equations|Maxwell's equations]]''' are a [[Mathematical descriptions of the electromagnetic field| formulation of Maxwell's equations ]] using [[matrix (mathematics)|matrices]], [[complex numbers]], and [[vector calculus]]. These representations are for a [[homogeneous medium]], an approximation in an [[inhomogeneous medium]]. A matrix representation for an inhomogeneous medium was presented using a pair of matrix equations.<ref>(Bialynicki-Birula, 1994, 1996a, 1996b)</ref> A single equation using 4 × 4 matrices is necessary and sufficient for any homogeneous medium. For an inhomogeneous medium it necessarily requires 8 × 8 matrices.<ref>(Khan, 2002, 2005)</ref>

==Introduction==

Maxwell's equations in the standard vector calculus formalism, in an inhomogeneous medium with sources, are:<ref>(Jackson, 1998; Panofsky and Phillips, 1962)</ref>

:<math>
\begin{align}
& {\mathbf \nabla} \cdot {\mathbf D} \left({\mathbf r} , t \right)
=
\rho\,, \\
& {\mathbf \nabla} \times {\mathbf H} \left({\mathbf r} , t \right)
- \frac{\partial }{\partial t}
{\mathbf D} \left({\mathbf r} , t \right)
=
{\mathbf J}\,, \\
& {\mathbf \nabla} \times {\mathbf E} \left({\mathbf r} , t \right)
+
\frac{\partial }{\partial t}
{\mathbf B} \left({\mathbf r} , t \right)
= 0\,, \\
& {\mathbf \nabla} \cdot {\mathbf B} \left({\mathbf r} , t \right)
= 0\,.
\end{align}
</math>

The media is assumed to be [[linear media|linear]], that is

:<math>{\mathbf D} = \epsilon {\mathbf E}\,,\quad {\mathbf B} = \mu {\mathbf H}</math>,

where ε = ε('''r''', ''t'') is the [[Permittivity|permittivity of the medium]] and μ = μ('''r''', ''t'') the [[Permeability (electromagnetism)|permeability of the medium]] (see [[constitutive equation]]). For a homogeneous medium ε and μ are constants.
The [[speed of light]] in the medium is given by
:<math>v ({\mathbf r} , t) = \frac{1}{\sqrt{\epsilon ({\mathbf r} , t) \mu ({\mathbf r} , t)}}</math>.

In vacuum, ε0 = 8.85 × 10−12 C2·N−1·m−2 and μ0 = 4π × 10−7 H·m−1

One possible way to obtain the required matrix representation is
to use the [[Riemann-Silberstein vector]] <ref>(Silberstein, 1907a, 1907b, Bialynicki-Birula, 1996b)</ref> given by

:<math>
\begin{align}
{\mathbf F}^{+} \left({\mathbf r} , t \right)
& =
\frac{1}{\sqrt{2}}
\left(
\sqrt{\epsilon ({\mathbf r} , t)} {\mathbf E} \left({\mathbf r} , t \right)
+ {\rm i} \frac{1}{\sqrt{\mu ({\mathbf r} , t)}} {\mathbf B} \left({\mathbf r} , t \right) \right) \\
{\mathbf F}^{-} \left({\mathbf r} , t \right)
& =
\frac{1}{\sqrt{2}}
\left(
\sqrt{\epsilon ({\mathbf r} , t)} {\mathbf E} \left({\mathbf r} , t \right)
- {\rm i} \frac{1}{\sqrt{\mu ({\mathbf r} , t)}} {\mathbf B} \left({\mathbf r} , t \right) \right)\,.
\end{align}
</math>

If for a certain medium ε = ε('''r''', ''t'') and μ = μ('''r''', ''t'') are constants (or can be treated as ''local'' constants under certain approximations), then the vectors '''F'''± ('''r''', ''t'') satisfy

:<math>
\begin{align}
{\rm i} \frac{\partial }{\partial t} {\mathbf F}^{\pm} \left({\mathbf r} , t \right)
& =
\pm v {\mathbf \nabla} \times {\mathbf F}^{\pm} \left({\mathbf r} , t \right)
- \frac{1}{\sqrt{2 \epsilon}} ({\rm i} {\mathbf J}) \\
{\mathbf \nabla} \cdot {\mathbf F}^{\pm} \left({\mathbf r} , t \right)
& =
\frac{1}{\sqrt{2 \epsilon}} (\rho)\,.
\end{align}
</math>

Thus by using the Riemann-Silberstein vector, it is possible to reexpress the Maxwell's equations for a
medium with constant ε = ε('''r''', ''t'') and μ = μ('''r''', ''t'') as a
pair of equations.

== Homogeneous medium ==
In order to obtain a single matrix equation instead of a pair, the following new functions
are constructed using the components of the Riemann-Silberstein vector<ref>(Khan, 2002, 2005)</ref>

:<MATH>\begin{align}
\Psi^{+} ({\mathbf r} , t)
& =
\left[
\begin{array}{c}
- F_x^{+} + {\rm i} F_y^{+} \\
F_z^{+} \\
F_z^{+} \\
F_x^{+} + {\rm i} F_y^{+}
\end{array}
\right]\,, \quad
\Psi^{-} ({\mathbf r} , t)
=
\left[
\begin{array}{c}
- F_x^{-} - {\rm i} F_y^{-} \\
F_z^{-} \\
F_z^{-} \\
F_x^{-} - {\rm i} F_y^{-}
\end{array}
\right]\,.
\end{align}</MATH>

The vectors for the sources are

:<MATH>\begin{align}
W^{+}
& =
\left(\frac{1}{\sqrt{2 \epsilon}}\right)
\left[
\begin{array}{c}
- J_x + {\rm i} J_y \\
J_z - v \rho \\
J_z + v \rho \\
J_x + {\rm i} J_y
\end{array}
\right]\,, \quad
W^{-}
=
\left(\frac{1}{\sqrt{2 \epsilon}}\right)
\left[
\begin{array}{c}
- J_x - {\rm i} J_y \\
J_z - v \rho \\
J_z + v \rho \\
J_x - {\rm i} J_y
\end{array}
\right]\,.
\end{align}</MATH>

Then,

:<MATH>\begin{align}
\frac{\partial}{\partial t}
\Psi^{+}
& =
- v
\left\{ {\mathbf M} \cdot {\mathbf \nabla} \right\} \Psi^{+}
- W^{+}\, \\
\frac{\partial}{\partial t}
\Psi^{-}
& =
- v
\left\{ {\mathbf M}^{*} \cdot {\mathbf \nabla} \right\} \Psi^{-}
- W^{-}\,,
\end{align}</MATH>

where * denotes [[complex conjugation]] and the triplet, '''M''' = (''Mx'', ''My'', ''Mz'') is expressed in terms of

:<MATH>
\Omega =
\begin{pmatrix}
{\mathbf 0} & - {\mathbf l} \\
{\mathbf l} & {\mathbf 0}
\end{pmatrix}
\,, \qquad
\beta =
\begin{pmatrix}
{\mathbf l} & {\mathbf 0} \\
{\mathbf 0} & - {\mathbf l}
\end{pmatrix}
\,, \qquad
{\mathbf l} =
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\,.</MATH>

Alternately, one may use the matrix ''J'' = −Ω. Both differ by a sign. For our purpose it is fine to use either Ω or ''J''. However, they have a different meaning: ''J'' is [[Covariance and contravariance of vectors|contravariant]]
and Ω is [[Covariance and contravariance of vectors|covariant]]. The matrix Ω corresponds to the [[Lagrange bracket]]s of [[classical mechanics]] and ''J'' corresponds to the [[Poisson bracket]]s. An important relation is Ω = ''J''−1. The ''M''-matrices are

:<MATH>
M_x =
\begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{bmatrix}
= - \beta \Omega\,,
</MATH>

:<MATH>
M_y =
\begin{bmatrix}
0 & 0 & - {\rm i} & 0 \\
0 & 0 & 0 & - {\rm i} \\
{\rm i} & 0 & 0 & 0 \\
0 & {\rm i} & 0 & 0
\end{bmatrix}
= {\rm i} \Omega\,,
</MATH>

:<MATH>M_z =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & - 1
\end{bmatrix}
= \beta \,.
</MATH>

Each of the four Maxwell's equations are obtained from the
matrix representation. This is done by taking
the sums and differences of row-I with row-IV and row-II with row-III
respectively. The first three give the ''y'', ''x'' and ''z'' components
of the [[Curl (mathematics)|curl]] and the last one gives the [[divergence]] conditions.

It is to be noted that the [[Matrix (mathematics)|matrices]] '''M''' are all [[Non-singular matrix|non-singular]] and all are [[Hermitian matrices|Hermitian]]. Moreover, they satisfy the usual algebra of the [[Dirac matrices]], including,

:<MATH>\begin{align}
M_x \beta = - \beta M_x\,, \\
M_y \beta = - \beta M_y\,, \\
M_x^2 = M_y^2 = M_z^2 = I\,, \\
M_x M_y = - M_y M_x = {\rm i} M_z\,, \\
M_y M_z = - M_z M_y = {\rm i} M_x\,, \\
M_z M_x = - M_x M_z = {\rm i} M_y\,.
\end{align}</MATH>

It is to be noted that the (Ψ±, '''M''') are ''not'' unique. Different choices of Ψ± would give rise to different '''M''', such that the triplet '''M''' continues to satisfy the algebra of the Dirac matrices. The Ψ± ''via'' the Riemann-Silberstein vector has certain advantages over the other possible choices.<ref>(Bialynicki-Birula, 1996b)</ref> The Riemann-Silberstein vector is well known in [[classical electrodynamics]] and has certain interesting properties and uses.<ref>(Bialynicki-Birula, 1996b)</ref>

In deriving the above 4 × 4 matrix representation of the Maxwell's equations, the spatial and temporal derivatives of ε('''r''', ''t'') and μ('''r''', ''t'') in the first two of the Maxwell's equations have been ignored. The ε and μ have been treated as ''local'' constants.

== Inhomogeneous medium ==
In an inhomogeneous medium, the spatial and temporal variations of ε = ε('''r''', ''t'') and μ = μ('''r''', ''t'') are not zero.
That is they are no longer ''local'' constant. Instead of using ε = ε('''r''', ''t'') and μ = μ('''r''', ''t''), it is advantageous to use the two derived ''laboratory functions'' namely the [[Electrical resistance and conductance|resistance function]] and the [[Velocity of light|velocity function]]

:<MATH>\begin{align}
\text{ Velocity function:} \, v ({\mathbf r} , t)
& =
\frac{1}{\sqrt{\epsilon ({\mathbf r} , t) \mu ({\mathbf r} , t)}} \\
\text{Resistance function:} \, h ({\mathbf r} , t)
& =
\sqrt{\frac{\mu ({\mathbf r} , t)}{\epsilon ({\mathbf r} , t)}}\,.
\end{align}</MATH>

In terms of these functions:
:<MATH>\varepsilon = \frac{1}{v h}\,,\quad \mu = \frac{h}{v}</MATH>.
These functions occur in the matrix representation through their [[logarithmic derivative]]s;

:<MATH>\begin{align}
{\mathbf u} ({\mathbf r} , t)
& =
\frac{1}{2 v ({\mathbf r} , t)} {\mathbf \nabla} v ({\mathbf r} , t)
=
\frac{1}{2} {\mathbf \nabla} \left\{\ln v ({\mathbf r} , t) \right\}
=
- \frac{1}{2} {\mathbf \nabla} \left\{\ln n ({\mathbf r} , t) \right\} \\
{\mathbf w} ({\mathbf r} , t)
& =
\frac{1}{2 h ({\mathbf r} , t)} {\mathbf \nabla} h ({\mathbf r} , t)
=
\frac{1}{2} {\mathbf \nabla} \left\{\ln h ({\mathbf r} , t) \right\}\,,
\end{align}</MATH>

where
:<MATH>n ({\mathbf r} , t) = \frac{c}{v ({\mathbf r} , t)}</MATH>

is the [[refractive index]] of the medium.

The following matrices naturally arise in the exact matrix representation of the Maxwell's equation in a medium

:<MATH>\begin{align}
{\mathbf \Sigma}
=
\left[
\begin{array}{cc}
{\mathbf \sigma} & {\mathbf 0} \\
{\mathbf 0} & {\mathbf \sigma}
\end{array}
\right]\,, \qquad
{\mathbf \alpha}
=
\left[
\begin{array}{cc}
{\mathbf 0} & {\mathbf \sigma} \\
{\mathbf \sigma} & {\mathbf 0}
\end{array}
\right]\,, \qquad
{\mathbf I}
=
\left[
\begin{array}{cc}
{\mathbf 1} & {\mathbf 0} \\
{\mathbf 0} & {\mathbf 1}
\end{array}
\right]\,,
\end{align}</MATH>

where '''Σ''' are the [[Gamma matrices|Dirac spin matrices]] and '''α''' are the matrices used in the [[Dirac equation]], and '''σ''' is the triplet of the [[Pauli matrices]]

:<MATH>
{\mathbf \sigma} = (\sigma_x , \sigma_y , \sigma_z)
=
\left[
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
,
\begin{pmatrix}
0 & - {\rm i} \\
{\rm i} & 0
\end{pmatrix}
,
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\right]</MATH>

Finally, the matrix representation is

:<MATH>\begin{align}
&
\frac{\partial }{\partial t}
\left[
\begin{array}{cc}
{\mathbf I} & {\mathbf 0} \\
{\mathbf 0} & {\mathbf I}
\end{array}
\right]
\left[
\begin{array}{cc}
\Psi^{+} \\
\Psi^{-}
\end{array}
\right]
-
\frac{\dot{v} ({\mathbf r} , t)}{2 v ({\mathbf r} , t)}
\left[
\begin{array}{cc}
{\mathbf I} & {\mathbf 0} \\
{\mathbf 0} & {\mathbf I}
\end{array}
\right]
\left[
\begin{array}{cc}
\Psi^{+} \\
\Psi^{-}
\end{array}
\right]
+ \frac{\dot{h} ({\mathbf r} , t)}{2 h ({\mathbf r} , t)}
\left[
\begin{array}{cc}
{\mathbf 0} & {\rm i} \beta \alpha_y \\
{\rm i} \beta \alpha_y & {\mathbf 0}
\end{array}
\right]
\left[
\begin{array}{cc}
\Psi^{+} \\
\Psi^{-}
\end{array}
\right] \\
& = - v ({\mathbf r} , t)
\left[
\begin{array}{ccc}
\left\{
{\mathbf M} \cdot {\mathbf \nabla}
+
{\mathbf \Sigma} \cdot {\mathbf u}
\right\}
& &
- {\rm i} \beta
\left({\mathbf \Sigma} \cdot {\mathbf w}\right)
\alpha_y
\\
- {\rm i} \beta
\left({\mathbf \Sigma}^{*} \cdot {\mathbf w}\right)
\alpha_y
&
\left\{
{\mathbf M}^{*} \cdot {\mathbf \nabla}
+
{\mathbf \Sigma}^{*} \cdot {\mathbf u}
\right\}
\end{array}
\right]
\left[
\begin{array}{cc}
\Psi^{+} \\
\Psi^{-}
\end{array}
\right]
- \left[
\begin{array}{cc}
{\mathbf I} & {\mathbf 0} \\
{\mathbf 0} & {\mathbf I}
\end{array}
\right]
\left[
\begin{array}{c}
W^{+} \\
W^{-}
\end{array}
\right]\,,
\end{align}</MATH>

The above representation contains thirteen 8 × 8 matrices. Ten of these are [[Hermitian matrices|Hermitian]]. The exceptional ones are the ones that contain the three components of '''w'''('''r''', ''t''), the logarithmic gradient of the resistance function. These three matrices, for the resistance function are [[antihermitian]].

The Maxwell's equations have been expressed in a matrix form for a medium with varying permittivity ε = ε('''r''', ''t'') and permeability μ = μ('''r''', ''t''), in presence of sources. This representation uses a single matrix equation, instead of a ''pair'' of matrix equations. In this representation, using 8 × 8 matrices, it has been possible to separate the dependence of the coupling between the upper components (Ψ+) and the lower components (Ψ−) through the two laboratory functions.
Moreover the exact matrix representation has an algebraic structure very similar to the Dirac equation.<ref>(Khan, 2002, 2005)</ref>
It is interesting to note that the Maxwell's equations can be derived from the [[Fermat's principle]] of [[geometrical optics]] by the process of "wavization"{{clarify|date=March 2013}} analogous to the [[Canonical quantization|quantization]] of [[classical mechanics]].<ref>(Pradhan, 1987)</ref>

== Applications ==

One of the early uses of the matrix forms of the Maxwell's equations was to study certain symmetries, and the similarities with the Dirac equation.

The matrix form of the Maxwell's equations is used as a candidate for the [[Riemann-Silberstein vector#Photon wave function|Photon Wavefunction]].<ref>(Bialynicki-Birula, 1996b)</ref>

Historically, the [[geometrical optics]] is based on the [[Fermats principle|Fermat’s principle of least time]]. Geometrical optics can be completely derived from the Maxwell's equations. This is traditionally done using the [[Helmholtz equation]]. It is to be noted that the derivation of the Helmholtz equation from the [[Maxwell equation|Maxwell’s equations]] is an approximation as one neglects the spatial and temporal derivatives of the permittivity and permeability of the medium. A new formalism of light beam optics has been developed, starting with the Maxwell’s equations in a matrix form: a single entity containing all the four Maxwell’s equations.
Such a prescription is sure to provide a deeper understanding of beam-optics and [[Photon polarization|polarization]] in a unified manner.<ref>(Khan, 2006b, 2010)</ref>
The beam-optical Hamiltonian derived from this matrix representation has an algebraic structure very similar to the [[Dirac equation]], making it emnable to the [[Foldy-Wouthuysen transformation|Foldy-Wouthuysen technique]].<ref>(Khan, 2006a, 2008)</ref> This approach is very similar to one developed for the quantum theory of charged-particle beam optics.<ref>(Jagannathan et al., 1989, Jagannathan, 1990, Jagannathan and Khan 1996, Khan, 1997)</ref>

== References ==


===Notes===

{{reflist}}

===Others===

{{Refbegin}}
* Bialynicki-Birula, I. (1994). On the wave function of the photon. Acta Physica Polonica A, '''86''', 97-116.

* Bialynicki-Birula, I. (1996a). The Photon Wave Function. In ''Coherence and Quantum Optics VII''. [[Joseph H. Eberly|Eberly, J. H.]], [[Leonard Mandel|Mandel, L.]] and [[Emil Wolf]] (ed.), Plenum Press, New York, 313.

* Bialynicki-Birula, I. (1996b). [http://arxiv.org/abs/quant-ph/0508202 Photon wave function]. in [[Progress in Optics]], Vol. XXXVI, [[Emil Wolf]]. (ed.), [[Elsevier]], Amsterdam, 245-294.

* [[John David Jackson (physicist)|Jackson, J. D.]] (1998). ''Classical Electrodynamics'', Third Edition, John Wiley & Sons.
* [http://scholar.google.com/citations?user=mp7XSDAAAAAJ&hl=en Jagannathan, R.], (1990). [http://dx.doi.org/10.1103/PhysRevA.42.6674 Quantum theory of electron lenses based on the Dirac equation]. ''Physical Review A'', '''42''', 6674-6689.
* [http://inspirehep.net/author/R.Jagannathan.1/ Jagannathan, R.] and [http://inspirehep.net/author/S.A.Khan.5/ Khan, S. A.] (1996). [http://dx.doi.org/10.1016/S1076-5670(08)70096-X Quantum theory of the optics of charged particles]. In Hawkes Peter, W. (ed.), ''Advances in Imaging and Electron Physics'', Vol. '''97''', Academic Press, San Diego, pp. 257–358.
* [http://scholar.google.com/citations?user=mp7XSDAAAAAJ&hl=en Jagannathan, R.], [[Rajiah Simon|Simon, R.]], [[George Sudarshan|Sudarshan, E. C. G.]] and [[Narasimhaiengar Mukunda|Mukunda, N.]] (1989). [http://dx.doi.org/10.1016/0375-9601(89)90685-3 Quantum theory of magnetic electron lenses based on the Dirac equation]. ''Physics Letters A'' '''134''', 457-464.
* [http://inspirehep.net/author/S.A.Khan.5/ Khan, S. A.] (1997). [http://www.imsc.res.in/xmlui/handle/123456789/75?show=full Quantum Theory of Charged-Particle Beam Optics], ''Ph.D Thesis'', [[University of Madras]], [[Chennai]], [[India]]. (complete thesis available from [http://www.imsc.res.in/xmlui/ Dspace of IMSc Library], [[Institute of Mathematical Sciences, Chennai|The Institute of Mathematical Sciences]], where the doctoral research was done).
* [http://arxiv.org/a/khan_s_1 Sameen Ahmed Khan]. (2002). [http://arXiv.org/abs/physics/0205083/ Maxwell Optics: I. An exact matrix representation of the Maxwell equations in a medium]. ''E-Print'': http://arXiv.org/abs/physics/0205083/.
* [http://inspirehep.net/author/S.A.Khan.5/ Sameen Ahmed Khan]. (2005). [http://dx.doi.org/10.1238/Physica.Regular.071a00440 An Exact Matrix Representation of Maxwell's Equations]. ''Physica Scripta'', '''71'''(5), 440-442.
* [http://scholar.google.com/citations?user=hZvL5eYAAAAJ&hl Sameen Ahmed Khan]. (2006a). [http://dx.doi.org/10.1016/j.ijleo.2005.11.010 The Foldy-Wouthuysen Transformation Technique in Optics]. ''Optik-International Journal for Light and Electron Optics''. '''117'''(10), pp. 481–488 http://www.elsevier-deutschland.de/ijleo/.
* [http://arxiv.org/a/khan_s_1 Sameen Ahmed Khan]. (2006b). Wavelength-Dependent Effects in Light Optics. in ''New Topics in Quantum Physics Research'', Editors: Volodymyr Krasnoholovets and Frank Columbus, [http://www.novapublishers.com/ Nova Science Publishers], New York, pp. 163–204. (ISBN 1600210287 and ISBN 978-1600210280).
* [http://arxiv.org/a/khan_s_1 Sameen Ahmed Khan]. (2008). [http://dx.doi.org/10.1016/S1076-5670(08)00602-2 The Foldy-Wouthuysen Transformation Technique in Optics], In Hawkes Peter, W. (ed.), ''Advances in Imaging and Electron Physics'', Vol. 152, [[Elsevier]], Amsterdam, pp. 49–78. (ISBN 0123742196 and ISBN 978-0-12-374219-3).
* [http://oeis.org/wiki/User:Sameen_Ahmed_Khan Sameen Ahmed Khan]. (2010). [http://dx.doi.org/10.1016/j.ijleo.2008.07.027 Maxwell Optics of Quasiparaxial Beams], ''Optik-International Journal for Light and Electron Optics'', '''121'''(5), 408-416. (http://www.elsevier-deutschland.de/ijleo/).

* [[Otto Laporte|Laporte, O.]], and [[George Uhlenbeck|Uhlenbeck, G. E.]] (1931). Applications of spinor analysis to the Maxwell and Dirac Equations. ''Physical Review'', '''37''', 1380-1397.

* [[Ettore Majorana|Majorana, E.]] (1974). (unpublished notes), quoted after Mignani, R., Recami, E., and Baldo, M. About a Diraclike Equation for the Photon, According to Ettore Majorana. ''Lett. Nuovo Cimento'', '''11''', 568-572.

* Moses, E. (1959).Solutions of Maxwell's equations in terms of a spinor notation: the direct and inverse problems. ''Physical Review'', '''113'''(6), 1670-1679.

* [[Wolfgang Panofsky|Panofsky, W. K. H.]], and [[Melba Phillips|Phillips, M.]] (1962). ''Classical Electricity and Magnetics'', Addison-Wesley Publishing Company, Reading, Massachusetts, USA.
* [http://inspirehep.net/author/T.Pradhan.1/ Pradhan, T]. (1987). [http://dx.doi.org/10.1016/0375-9601(87)90735-3 Maxwell's Equations From Geometrical Optics]. IP/BBSR/87-15; ''Physics Letters A'' '''122'''(8), 397-398.

* [[Ludwig Silberstein]]. (1907a). [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/silberstein_-_em_equations_in_bivector_fomr.pdf Elektromagnetische Grundgleichungen in bivektorieller Behandlung], Ann. Phys. (Leipzig), 22, 579-586.

* [[Ludwig Silberstein]]. (1907b). [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/silberstein_-_addendum.pdf Nachtrag zur Abhandlung ber Elektromagnetische Grundgleichungen in bivektorieller Behandlung]. Ann. Phys. (Leipzig), 24, 783-784.

{{Refend}}

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en>Jonesey95: Fixing "Pages with citations using unnamed parameters" error.