View source for Maxwell equations

The equations of an electromagnetic field in material surroundings; established in the 1860's by J.C. Maxwell on the basis of the experimental evidence at that time of the laws of electric and magnetic phenomena.

In classical electrodynamics the electromagnetic field in a medium is described by four vector fields; the electric field strength <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631401.png" />, the electric displacement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631402.png" />, the magnetic field strength <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631403.png" />, and the magnetic flux density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631404.png" />. These are continuously-differentiable functions of the radius vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631405.png" /> of a point of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631406.png" />-dimensional space and of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631407.png" />.

The Maxwell equations are a system of inhomogeneous partial differential equations of the first order for the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m0631409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314011.png" />, which, in SI-units, takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1a)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1b)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1c)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1d)</td></tr></table>

where the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314016.png" /> — a given volume density of electric charge in the medium — and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314017.png" /> — the volume density of electric current (the charge passing in unit time through a unit area perpendicular to the direction of motion of the charge) — are the sources of the flow. The Maxwell equations may also be described in integral form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314019.png" /> is any two-sided surface with closed boundary curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314021.png" /> is the closed boundary surface of any bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314023.png" />; further, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314024.png" /> is the elementary arc length along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314026.png" /> is the elementary oriented surface area, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314027.png" /> is the elementary volume on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314028.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314031.png" /> form a right-hand system, while on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314032.png" /> the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314033.png" /> points away from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314034.png" />.

The fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314039.png" /> are not independent. In a large number of material media, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314041.png" /> depend only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314042.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314043.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314044.png" />, that is, the following functional dependencies hold:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

called the equations of state or the constitutive equations of the medium. Within the limits of classical macroscopic electrodynamics the equations of state (3) must be given in addition (postulated or determined from experimental data) and then the equations for the two remaining independent vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314047.png" /> become closed. The concrete form of the equations of state (3) is determined by the electric and magnetic properties of the given medium and its states. In general, in (3), the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314050.png" />, at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314051.png" /> at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314052.png" />, may depend non-linearly on the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314054.png" />, respectively, at all points of the medium (non-local case) and at all times prior to, by the physical principle of causality, the given time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314055.png" /> (a medium with after-effects or memory). Most media of practical interest are characterized by local linear dependency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314060.png" />, and, in this case, the Maxwell equations turn out to be linear; however, in applications, more complicated cases are met (for example, in non-linear optics).

The equations of states (3) may be deduced, in principle, from microscopic electrodynamics and taking account of the motions of different parts of the medium, their individual microscopic characteristics (the values of electrical charge, mass) and their interactions. The values of the macroscopic fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314063.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314064.png" /> are then defined as the volume-average values of the microscopic fields created by the individual motions of the charged particles in the medium and for them the Maxwell equations hold.

On the boundary surface between two different media the following boundary conditions must be satisfied:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314065.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314066.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314067.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314068.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314069.png" /> is the surface current density, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314070.png" /> is the surface charge density, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314071.png" /> is a unit vector normal to the boundary surface, and the subscripts 1 and 2 refer to the values of the fields on either side of the boundary.

A consequence of the Maxwell equations is the continuity equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314072.png" /></td> </tr></table>

expressing the law of conservation of electrical charge.

The Maxwell equations are invariant under [[Lorentz transformation|Lorentz transformation]]. In the pseudo-Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314073.png" />-dimensional space-time with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314076.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314077.png" /> one introduces two anti-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314078.png" />-dimensional tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314080.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314081.png" />) with components

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314082.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314083.png" />-dimensional current vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314084.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314085.png" />, whose spatial components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314086.png" /> coincide with the components of the current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314087.png" /> and whose fourth component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314088.png" /> is proportional to the charge density, then the Maxwell equations

may be written in relativistic covariant form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

In these equations, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314091.png" /> is the speed of light in vacuum. The equations (5) are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314092.png" />-dimensional form of (1b) and (1c), and the equations (6) are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314093.png" />-dimensional form of (1a) and (1d).

For an electromagnetic field in vacuum, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314095.png" />, and consequently <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314096.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314097.png" /> is the permeability of vacuum and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314098.png" /> is the permitivity of vacuum, and the electromagnetic field is described by just one tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m06314099.png" />. If one introduces a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140100.png" />-dimensional electromagnetic vector potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140102.png" />, whose spatial components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140103.png" /> form the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140104.png" />-dimensional vector potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140105.png" /> and whose fourth time component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140106.png" /> is proportional to the scalar potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140107.png" />, then the components of the anti-symmetric electromagnetic field tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140108.png" /> can be expressed in terms of the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140109.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140110.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

Because of (7), the equations (5) are satisfied identically and the equations (6) take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140111.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

that is, are inhomogeneous wave equations for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140112.png" />. The introduction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140113.png" /> allows the Maxwell equations to be written in the simple form (8). However, the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140114.png" /> is not uniquely defined, which reflects the invariance of the Maxwell equations in the form (8) relative to gauge transformations. This non-uniqueness in the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140115.png" /> can be removed (see [[Gauge transformation|Gauge transformation]]).

According to (4) and (7) the physically observable fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140117.png" /> can be expressed in terms of the vector potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140118.png" /> and the scalar potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140119.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140120.png" /></td> </tr></table>

When the electromagnetic field in vacuum is free from sources, the Maxwell equations

and (8) become homogeneous and it is possible to obtain from them homogeneous wave equations for the electric and magnetic field strength:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140121.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140122.png" /> is the Laplacian (cf. [[Laplace operator|Laplace operator]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063140/m063140123.png" /> is the speed of propagation of electromagnetic waves in vacuum.

The Maxwell equations for an electromagnetic field are used only in the classical theory. Thus, when the variable electric and magnetic field have very high frequencies and very small wavelengths (comparable with the dimensions of the atoms), significant quantum effects arise and the theory of an electromagnetic field and its sources must be built on the basis of quantum electrodynamics.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.C. Maxwell,   "A treatise on electricity and magnetism" , Clarendon Press  (1873)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.E. Tamm,   "Fundamentals of the theory of electricity" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "The classical theory of fields" , Pergamon  (1951)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "Electrodynamics of continous media" , Pergamon  (1960)  (Translated from Russian)</TD></TR></table>



====Comments====
For a historical survey see [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Jackson,   "Classical electrodynamics" , Wiley  (1962)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Statton,   "Electromagnetic theory" , McGraw-Hill  (1941)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.T. Whittaker,   "A history of the theories of aether and electricity" , '''1–2''' , Nelson  (1951–1953)</TD></TR></table>