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 $ \mathbf E $, the electric displacement $ \mathbf D $, the magnetic field strength $ \mathbf H $, and the magnetic flux density $ \mathbf B $. These are continuously-differentiable functions of the radius vector $ \mathbf r $ of a point of the $ 3 $- dimensional space and of the time $ t $.

The Maxwell equations are a system of inhomogeneous partial differential equations of the first order for the fields $ \mathbf E $, $ \mathbf D $, $ \mathbf H $, and $ \mathbf B $, which, in SI-units, takes the form

$$ \tag{1a } - \frac{\partial \mathbf D }{\partial t } + \mathop{\rm rot} \ \mathbf H = \ \mathbf J , $$

$$ \tag{1b } \frac{\partial \mathbf B }{\partial t } + \mathop{\rm rot} \mathbf E = 0 , $$

$$ \tag{1c } \mathop{\rm div} \mathbf B = 0 , $$

$$ \tag{1d } \mathop{\rm div} \mathbf D = \rho , $$

where the terms $ \rho ( t , \mathbf r ) $ — a given volume density of electric charge in the medium — and $ \mathbf J ( t , \mathbf r ) $ — 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:

$$ \tag{2 } \left . \begin{array}{c} \oint _ { C } \mathbf H d \mathbf l = \int\limits _ { S } \left ( \mathbf J + \frac{\partial \mathbf D }{\partial t } \right ) d \mathbf s , \\ \oint _ { C } \mathbf E d \mathbf l = - \int\limits _ { S } \frac{\partial \mathbf B }{\partial t } d \mathbf s , \\ \oint _ {\partial V } \mathbf B d \mathbf s = 0 , \\ \oint _ {\partial V } \mathbf D d \mathbf s = \ 4 \pi \int\limits _ { V } \rho d V , \end{array} \right \} $$

where $ S $ is any two-sided surface with closed boundary curve $ C $ and $ \partial V $ is the closed boundary surface of any bounded domain $ C $ in $ \mathbf R ^ {3} $; further, $ d \mathbf l $ is the elementary arc length along $ C $, $ d \mathbf s $ is the elementary oriented surface area, and $ \partial V $ is the elementary volume on $ S $; $ C $, $ d \mathbf s $ and $ d \mathbf l $ form a right-hand system, while on $ \partial V $ the element $ d \mathbf s $ points away from $ V $.

The fields $ \mathbf E $, $ \mathbf D $, $ \mathbf H $, $ \mathbf B $, and $ \mathbf J $ are not independent. In a large number of material media, $ \mathbf D $ and $ \mathbf J $ depend only on $ \mathbf E $, and $ \mathbf B $ depends only on $ \mathbf H $, that is, the following functional dependencies hold:

$$ \tag{3 } \mathbf D = \mathbf D ( \mathbf E ) ,\ \ \mathbf J = \mathbf J ( \mathbf E ) ,\ \ \mathbf B = \mathbf B ( \mathbf H ) , $$

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 $ \mathbf E $ and $ \mathbf H $ 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 $ \mathbf D $, $ \mathbf J $ and $ \mathbf B $, at a point $ \mathbf r $ at time $ t $, may depend non-linearly on the values of $ \mathbf E $ and $ \mathbf H $, 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 $ t $( a medium with after-effects or memory). Most media of practical interest are characterized by local linear dependency of $ \mathbf D $ and $ \mathbf J $ on $ \mathbf E $, and $ \mathbf B $ on $ \mathbf H $, 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 $ \mathbf E $, $ \mathbf H $, $ \mathbf D $, and $ \mathbf B $ 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:

$$ [ \mathbf n \times \mathbf H _ {2} ] - [ \mathbf n \times \mathbf H _ {1} ] = \ \mathbf J _ {S} , $$

$$ [ \mathbf n \times \mathbf E _ {2} ] - [ \mathbf n \times \mathbf E _ {1} ] = 0 , $$

$$ ( \mathbf n \mathbf D _ {2} ) - ( \mathbf n \mathbf D _ {1} ) = \sigma , $$

$$ ( \mathbf n \mathbf B _ {2} ) - ( \mathbf n \mathbf B _ {1} ) = 0 , $$

where $ \mathbf J _ {S} $ is the surface current density, $ \sigma $ is the surface charge density, $ \mathbf n $ 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

$$ \frac{\partial \rho }{\partial t } + \mathop{\rm div} \ \mathbf J = 0 , $$

expressing the law of conservation of electrical charge.

The Maxwell equations are invariant under Lorentz transformation. In the pseudo-Euclidean $ 4 $- dimensional space-time with coordinates $ x _ {1} = x $, $ x _ {2} = y $, $ x _ {3} = z $, and $ x _ {4} = ict $ one introduces two anti-symmetric $ 4 $- dimensional tensors $ F _ {kl} $ and $ G _ {kl} $( $ k , l = 1 , 2 , 3 , 4 $) with components

$$ \tag{4 } \left . \begin{array}{c} F _ {12} = B _ {z} ,\ \ F _ {23} = B _ {x} ,\ \ F _ {31} = B _ {y} ,\ \ F _ {4k} = \frac{i}{c} E _ {k} , \\ G _ {12} = H _ {z} ,\ \ G _ {23} = H _ {x} ,\ \ G _ {31} = H _ {y} ,\ \ G _ {4k} = i c D _ {k} , \\ k = 1, 2, 3, 4, \\ \end{array} \right \} $$

and a $ 4 $- dimensional current vector $ j _ {k} $ $ ( k = 1 , 2 , 3 , 4 ) $, whose spatial components $ j _ {1} = j _ {x} , j _ {2} = j _ {y} , j _ {3} = j _ {z} $ coincide with the components of the current $ \mathbf j $ and whose fourth component $ j _ {4} = ic \rho $ is proportional to the charge density, then the Maxwell equations

may be written in relativistic covariant form:

$$ \tag{5 } \frac{\partial F _ {kl} }{\partial x _ {m} } + \frac{\partial F _ {lm} }{\partial x _ {k} } + \frac{\partial F _ {mk} }{\partial x _ {l} } = 0 ,\ \ k , l , m = 1 , 2 , 3 , 4 , $$

and

$$ \tag{6 } \sum_{l=1}^ { 4 } \frac{\partial G _ {kl} }{\partial x _ {l} } = \ j _ {k} ,\ \ k = 1 , 2 , 3 , 4 . $$

In these equations, $ c = 299 792 458 m/s $ is the speed of light in vacuum. The equations (5) are the $ 4 $- dimensional form of (1b) and (1c), and the equations (6) are the $ 4 $- dimensional form of (1a) and (1d).

For an electromagnetic field in vacuum, $ \mathbf D \equiv \epsilon _ {0} \mathbf E $ and $ \mathbf B \equiv \mu _ {0} \mathbf H $, and consequently $ G _ {kl} = F _ {kl} / \mu _ {0} $, where $ \mu _ {0} = 4 \pi \cdot 10 ^ {-7} H / m $ is the permeability of vacuum and $ \epsilon _ {0} = 1 / \mu _ {0} c ^ {2} $ is the permitivity of vacuum, and the electromagnetic field is described by just one tensor $ F _ {kl} $. If one introduces a $ 4 $- dimensional electromagnetic vector potential $ A _ {k} $, $ k = 1 , 2 , 3 , 4 $, whose spatial components $ A _ {1} = A _ {x} , A _ {2} = A _ {y} , A _ {3} = A _ {z} $ form the so-called $ 3 $- dimensional vector potential $ \mathbf A ( t , \mathbf r ) $ and whose fourth time component $ A _ {4} = ( i / c ) \phi $ is proportional to the scalar potential $ \phi ( t , \mathbf r ) $, then the components of the anti-symmetric electromagnetic field tensor $ F _ {kl} $ can be expressed in terms of the components $ A _ {k} $ by

$$ \tag{7 } F _ {kl} = \ \frac{\partial A _ {k} }{\partial x _ {l} } - \frac{\partial A _ {l} }{\partial x _ {k} } ,\ \ k , l = 1 , 2 , 3 , 4 . $$

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

$$ \tag{8 } \sum_{l=1}^ { 4 } \left ( \frac{\partial ^ {2} A _ {k} }{\partial x _ {l} ^ {2} } - \frac \partial {\partial x _ {k} } \frac{\partial A _ {l} }{\partial x _ {l} } \right ) = - \mu _ {0} j _ {k} ,\ \ k = 1 , 2 , 3 , 4 , $$

that is, are inhomogeneous wave equations for the $ A _ {k} $. The introduction of $ A _ {k} $ allows the Maxwell equations to be written in the simple form (8). However, the potential $ A _ {k} $ 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 $ A _ {k} $ can be removed (see Gauge transformation).

According to (4) and (7) the physically observable fields $ \mathbf E $ and $ \mathbf H $ can be expressed in terms of the vector potential $ \mathbf A $ and the scalar potential $ \phi $:

$$ \mathbf H = \frac{1}{\mu _ {0} } \mathop{\rm rot} \mathbf A ,\ \ \mathbf E = - \mathop{\rm grad} \phi - \frac{\partial \mathbf A }{\partial t } . $$

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:

$$ \frac{1}{c ^ {2} } \frac{\partial ^ {2} }{\partial t ^ {2} } \mathbf E - \Delta \mathbf E = 0 ,\ \ \frac{1}{c ^ {2} } \frac{\partial ^ {2} }{\partial t ^ {2} } \mathbf H - \Delta \mathbf H = 0 , $$

where $ \Delta $ is the Laplacian (cf. Laplace operator) and $ c $ 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.

Comments

For a historical survey see [a3].

References

[1]	J.C. Maxwell, "A treatise on electricity and magnetism" , Clarendon Press (1873) Zbl 05.0556.01
[2]	I.E. Tamm, "Fundamentals of the theory of electricity" , MIR (1979) (Translated from Russian)
[3]	L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Pergamon (1951) (Translated from Russian)
[4]	L.D. Landau, E.M. Lifshitz, "Electrodynamics of continous media" , Pergamon (1960) (Translated from Russian)
[a1]	J.D. Jackson, "Classical electrodynamics" , Wiley (1962)
[a2]	J.A. Statton, "Electromagnetic theory" , McGraw-Hill (1941)
[a3]	E.T. Whittaker, "A history of the theories of aether and electricity" , 1–2 , Nelson (1951–1953)