Maxwell's equations

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James Clark Maxwell found in 1868 four equations to describe all phenomena of electromagnetism:

1 Maxwell's Equations in the classical forms
2 The meaning of the equations
3 Covariant Formulation

[change] Maxwell's Equations in the classical forms

Name	Differential form	Integral form
Gauss' law:	$\nabla \cdot \mathbf{D} = \rho$	$\oint_S \mathbf{D} \cdot d\mathbf{A} = \int_V \rho \cdot dV$
Gauss' law for magnetism (absence of magnetic monopoles):	$\nabla \cdot \mathbf{B} = 0$	$\oint_S \mathbf{B} \cdot d\mathbf{A} = 0$
Faraday's law of induction:	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$	$\oint_C \mathbf{E} \cdot d\mathbf{l} - \oint_C \mathbf {B} \times \mathbf{v} \cdot d{\mathbf {l}} = - \ { d \over dt } \int_S \mathbf{B} \cdot d\mathbf{A}$
Ampère's law (with Maxwell's extension):	$\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}$	$\oint_C \mathbf{H} \cdot d\mathbf{l} = \int_S \mathbf{J} \cdot d \mathbf{A} + \int_S\frac{\partial \mathbf{D} }{\partial t}\cdot d \mathbf{A}$

where the following table provides the meaning of each symbol and the SI unit of measure:

Symbol	Meaning	SI Unit of Measure
$\mathbf{E}$	electric field	volt per metre
$\mathbf{H}$	magnetic field strength	ampere per metre
$\mathbf{D}$	electric displacement field	coulomb per square metre
$\mathbf{B}$	magnetic flux density also called the magnetic induction.	tesla, or equivalently, weber per square metre
$\ \rho \$	free electric charge density, not including dipole charges bound in a material	coulomb per cubic metre
$\mathbf{J}$	free current density, not including polarization or magnetization currents bound in a material	ampere per square metre
$d\mathbf{A}$	differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S	square meters
$dV \$	differential element of volume V enclosed by surface S	cubic meters
$d \mathbf{l}$	differential vector element of path length tangential to contour C enclosing surface c	meters
$\mathbf{v}$	instantaneous velocity of the line element $d\mathbf{l}$ defined above (for moving circuits).	meters per second

and

$\nabla \cdot$ is the divergence operator (SI unit: 1 per metre),

$\nabla \times$ is the curl operator (SI unit: 1 per metre).

[change] The meaning of the equations

[change] Charge density and the electric field

$\nabla \cdot \mathbf{D} = \rho$ ,

where $ρ$ is the free electric charge density (in units of C/m³), not including dipole charges bound in a material, and $\mathbf{D}$ is the electric displacement field (in units of C/m²). This equation corresponds to Coulomb's law for stationary charges in vacuum.

The equivalent integral form (by the divergence theorem), also known as Gauss' law, is:

$\oint_A \mathbf{D} \cdot d\mathbf{A} = Q_\mbox{enclosed}$

where $d\mathbf{A}$ is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and $Q enclosed$ is the free charge enclosed by the surface.

In a linear material, $\mathbf{D}$ is directly related to the electric field $\mathbf{E}$ via a material-dependent constant called the permittivity, $ε$ :

$\mathbf{D} = \varepsilon \mathbf{E}$ .

Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as $ε 0$ , and appears in:

$\nabla \cdot \mathbf{E} = \frac{\rho_t}{\varepsilon_0}$

where, again, $\mathbf{E}$ is the electric field (in units of V/m), $ρ t$ is the total charge density (including bound charges), and $ε 0$ (approximately 8.854 pF/m) is the permittivity of free space. $ε$ can also be written as $\varepsilon_0 \cdot \varepsilon_r$ , where $ε r$ is the material's relative permittivity or its dielectric constant.

Compare Poisson's equation.

[change] The structure of the magnetic field

$\nabla \cdot \mathbf{B} = 0$

$\mathbf{B}$ is the magnetic flux density (in units of teslas, T), also called the magnetic induction.

Equivalent integral form:

$\oint_A \mathbf{B} \cdot d\mathbf{A} = 0$

$d\mathbf{A}$ is the area of a differential square on the surface $A$ with an outward facing surface normal defining its direction.

This equation only works if the integral is done over a closed surface. This equation says, that in every volume the sum of the incoming magnetic field lines equals the sum of the outgoing magnetical field lines. This means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere. Mathematically formulated: "There are no magnetic monopoles".

[change] A changing magnetic flux and the electric field

$\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}$

Equivalent integral Form:

$\oint_{s} \mathbf{E} \cdot d\mathbf{s} - \oint_s \mathbf{B} \times \mathbf {v} \cdot d \mathbf{l} = - \frac {d\Phi_{\mathbf{B}}} {dt}$ where $\Phi_{\mathbf{B}} = \int_{A} \mathbf{B} \cdot d\mathbf{A}$

where

Φ_B is the magnetic flux through the area A described by the second equation

E is the electric field generated by the magnetic flux

s is a closed path in which current is induced, such as a wire

v is the instantaneous velocity of the line element (for moving circuits).

The electromotive force (sometimes denoted $\mathcal{E}$ , not to be confused with the permittivity above) is equal to the value of this integral.

This law corresponds to the Faraday's law of electromagnetic induction.

Some textbooks show the right hand sign of the Integral form with an N (representing the number of coils of wire that are around the edge of A) in front of the flux derivative. The N can be taken care of in calculating A (multiple wire coils means multiple surfaces for the flux to go through), and it is an engineering detail so it has been omitted here.

The negative sign is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's law.

This equation relates the electric and magnetic fields. This equation e.g. describes how electric motors and electric generators work. In a motor or generator, the fixed excitation is provided by the field circuit and the varying voltage is measured across the armature circuit. Maxwell's equations apply to a right-handed coordinate system. To apply them unmodified to a left handed system would mean a reversal of polarity of magnetic fields (not inconsistent, but confusingly against convention).

[change] The source of the magnetic field

$\nabla \times \mathbf{H} = \mathbf{J} + \frac {\partial \mathbf{D}} {\partial t}$

where H is the magnetic field strength (in units of A/m), related to the magnetic flux B by a constant called the permeability, μ (B = μH), and J is the current density, defined by: J = ∫ρ_qvdV where v is a vector field called the drift velocity that describes the velocities of that charge carriers which have a density described by the scalar function ρ_q.

In free space, the permeability μ is the permeability of free space, μ₀, which is defined to be exactly 4π×10^-7 W/A·m. Also, the permittivity becomes the permittivity of free space ε₀. Thus, in free space, the equation becomes:

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

Equivalent integral form:

$\oint_s \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_\mbox{encircled} + \mu_0\varepsilon_0 \int_A \frac{\partial \mathbf{E}}{\partial t} \cdot d \mathbf{A}$

s is the edge of the open surface A (any surface with the curve s as its edge will do), and I_encircled is the current encircled by the curve s (the current through any surface is defined by the equation: I_{through A} = ∫_AJ·dA).

If the electric flux density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's law.

[change] Covariant Formulation

There are only two covariant Maxwell Equations, because the covariant field vector includes the electrical and the magnetical field.

Mathematical note: In this section the abstract index notation will be used.

In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form:

$J^ b = \partial_a F^{ab} \,\!$ ,

and

$0 = \partial_c F_{ab} + \partial_b F_{ca} + \partial_a F_{bc}$

the last of which is equivalent to:

$0 = \epsilon_{dabc}\partial^a F^{bc} \,\!$

where $\, J^a$ is the 4-current, $\, F^{ab}$ is the field strength tensor (written as a 4 × 4 matrix), $\, \epsilon_{abcd}$ is the Levi-Civita symbol, and $\partial_a = (\partial/\partial ct, \nabla)$ is the 4-gradient (so that $\partial_a \partial^a$ is the d'Alembertian operator). (The $a$ in the first equation is implicitly summed over, according to Einstein notation.) The first tensor equation expresses the two inhomogeneous Maxwell's equations: Gauss' law and Ampere's law with Maxwell's correction. The second equation expresses the other two, homogenous equations: Faraday's law of induction and the absence of magnetic monopoles.

More explicitly, $J^a = \, (c \rho, \vec J)$ (as a contravariant vector), in terms of the charge density ρ and the current density $\vec J$ . The 4-current satisfies the continuity equation

$J^a{}_{,a} \, = 0$

In terms of the 4-potential (as a contravariant vector) $A^{a} = \left(\phi, \vec A c \right)$ , where φ is the electric potential and $\vec A$ is the magnetic vector potential in the Lorentz gauge $\left ( \partial_a A^a = 0 \right )$ , F can be expressed as:

$F^{ab} = \partial^b A^a - \partial^a A^b \,\!$

which leads to the 4 × 4 matrix rank-2 tensor:

$F^{ab} = \left( \begin{matrix} 0 & -\frac {E_x}{c} & -\frac {E_y}{c} & -\frac {E_z}{c} \\ \frac{E_x}{c} & 0 & -B_z & B_y \\ \frac{E_y}{c} & B_z & 0 & -B_x \\ \frac{E_z}{c} & -B_y & B_x & 0 \end{matrix} \right) .$

The fact that both electric and magnetic fields are combined into a single tensor expresses the fact that, according to relativity, both of these are different aspects of the same thing—by changing frames of reference, what seemed to be an electric field in one frame can appear as a magnetic field in another frame, and vice versa.

Using the tensor form of Maxwell's equations, the first equation implies

$\Box F^{ab} = 0$ (See Electromagnetic four-potential for the relationship between the d'Alembertian of the four-potential and the four-current, expressed in terms of the older vector operator notation).

Different authors sometimes employ different sign conventions for the above tensors and 4-vectors (which does not affect the physical interpretation).

$\, F^{ab}$ and $\, F_{ab}$ are not the same: they are related by the Minkowski metric tensor $η$ : $F_{ab} =\, \eta_{ac} \eta_{bd} F^{cd}$ . This introduces sign changes in some of F's components; more complex metric dualities are encountered in general relativity.