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Electromagnetism

Electricity · Magnetism

Electrostatics
Electric charge · Coulomb\'s law Electric field · Gauss\' law Electric potential Electric dipole moment

Magnetostatics
Ampère\'s law · Electric current Magnetic field · Magnetic flux Biot-Savart law Magnetic dipole moment

Electrodynamics
Free space Lorentz force law EMF · Electromagnetic induction Faraday\'s law · Displacement current Maxwell\'s equations · EM field Electromagnetic radiation Eddy current · Maxwell tensor

Covariant formulation
Liénard-Wiechert Potentials Electromagnetic tensor EM Stress-energy tensor

Scientists
Coulomb · Ampere · Maxwell · others

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The formulation of Maxwell\'s equations in special relativity refers to ways of writing Maxwell\'s equations of electromagnetism in the formalism of special relativity. In order to more clearly express the fact that Maxwell\'s equations (in free space) 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 (cgs units).

The equations

Maxwell\'s equations can be written as two tensor equations

\frac{\partial F^{\alpha\beta}}{\partial x^\alpha}=\mu_0J^\beta

\qquad\hbox{and}\qquad 0=\epsilon^{\alpha\beta\gamma\delta}\frac{\partial F_{\alpha\beta}}{\partial x^\gamma}

where F ^αβ is the field strength tensor (which incorporates the electric and magnetic fields), see below, J ^α is the 4-current (incorporating both charge density and current density), see below, є ^αβγδ is the Levi-Civita symbol (a mathematical construct), μ₀ is the magnetic constant, and the indices behave according to the Einstein summation convention.

The first tensor equation is an expression of the two inhomogeneous Maxwell\'s equations, Gauss\'s Law and Ampere\'s Law (with Maxwell\'s correction). The second equation is an expression of the homogenous equations, Faraday\'s law of induction and the absence of magnetic monopoles.

Other notation

Without the summation convention or the Levi-Civita symbol, the equations would be written

\sum_{\alpha=ct,x,y,z}{\partial F^{\alpha\beta}\over\partial x^\alpha}=\mu_0J^\beta

\qquad\hbox{and}\qquad 0={\partial F_{\alpha\beta}\over\partial x^\gamma}

+{\partial F_{\beta\gamma}\over\partial x^\alpha}
+{\partial F_{\gamma\alpha}\over\partial x^\beta}

where all indices range from 0 to 3 (or, more descriptively, over the set {ct,x,y,z}), where c is the speed of light in free space and μ₀ is the magnetic constant. The first tensor equation corresponds to four scalar equations, one for each value of

\beta

. The second tensor equation actually corresponds to

4^3=64

different scalar equations, but only four of these are independent.

For convenience, professionals often write the 4-gradient (that is, the derivative with respect to x) using abbreviated notations; for instance,

{\partial F^{\alpha\beta}\over \partial x^\gamma}\equiv \partial_\gamma F^{\alpha\beta}\equiv {F^{\alpha\beta}}_{,\gamma}

Using the latter notation, Maxwell\'s equations can be written as ${F^{\alpha\beta}}_{,\alpha}=\mu_0 J^\beta$ and $\epsilon^{\alpha\beta\gamma\delta} {F_{\alpha\beta,\gamma}}=0\ .$

Charge conservation

The 4-current is a contravariant vector given by:

J^{\alpha}  = \,  (c \rho, \vec{J} )

where $\rho$ is the charge density and $\vec{J}$ is the current density.

The 4-current satisfies the continuity equation

J^{\alpha}_{,\alpha} \, \ \stackrel{\mathrm{def}}{=}\  \partial_{\alpha} J^{\alpha} \, = 0

Field strength tensor and the 4-potential

The field strength tensor, an antisymmetric tensor, can be written:

F^{\alpha\beta} = \partial^{\alpha} A^{\beta} - \partial^{\beta} A^{\alpha} \,\!

where

A^{\alpha} =  \left(\frac{\phi}{c}, \vec{A}  \right)

is the 4-potential, φ is the scalar potential and $\vec{A}$ is the vector potential.

When using metric (-+++), the field strength tensor is written in terms of fields as:

F^{\alpha\beta} = \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. Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett, p. 395. ISBN 0-7637-3827-1.

Maxwell\'s equations, in the absence of sources, reduce to a wave equation in the field strength:

\partial_{\gamma} \partial^{\gamma}  F^{\alpha\beta} \ \stackrel{\mathrm{def}}{=}\  \Box F^{\alpha\beta} \ \stackrel{\mathrm{def}}{=}\  \nabla^2 F^{\alpha\beta} - {1 \over c^2 } { \partial^2 F^{\alpha\beta} \over {\partial t }^2   }= 0

Here, $\partial_{\alpha} \partial^{\alpha}$ is the d\'Alembertian operator.

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

The covariant version of the field strength tensor $\, F_{ab}$ is related to contravariant version $\, F^{ab}$ by the Minkowski metric tensor $\eta$

F_{\alpha\beta} =\,  \eta_{\alpha\gamma} \eta_{\beta\delta} F^{\gamma\delta} = F^{\alpha\beta}

Lorentz force

Main article: Lorentz force

Fields are detected by their effect on the motion of matter. Electromagnetic fields affect the motion of particles through the Lorentz force. Using the Lorentz force, Newton\'s law of motion can be written in relativistic form using the field strength tensor asThe assumption is made that no forces other than those originating in E and B are present, for example, no gravitational or electroweak forces.

m c { d u^{\alpha} \over { d \tau }   } =  { {} \over {}    }F^{\alpha \beta} q u_{\beta}

where m is the particle mass, q is the charge, and

u_{\beta} = \eta_{\beta \alpha } u^{\alpha } = \eta_{\beta \alpha } { d x^{\alpha } \over {d \tau}   }

is the 4-velocity of the particle. Here, $\tau$ is c times the proper time of the particle.

The relativistic version of Newton\'s law of motion differs from its nonrelativistic counterpart. The relativistic version emerges in order to maintain consistency of the transformation of forces as required by Maxwell\'s equations. Einstein used the well known problem of moving magnets and conductors to motivate the changes to the Lorentz force. See Einstein On the electrodynamics of moving bodies §6

Lagrangian for classical electrodynamics

The Lagrangian for classical electrodynamics (in SI) is

\mathcal{L} = \mathcal{L}_{\mathrm{field}} + \mathcal{L}_{\mathrm{int}} = -\frac{1}{4 \mu_0} F^{\alpha\beta} F_{\alpha\beta} - J^{\alpha}A_{\alpha}

Electromagnetic stress-energy tensor

The electromagnetic stress-energy tensor is related to the field strength tensor by:

{ T^{\alpha \beta } }_{,\beta} =  { {} \over {}    }F^{\alpha \beta} J_{\beta}

{ T_{\alpha \beta } } =  { 1 \over { 4 \pi } } \left (   F_{\alpha \mu} {F_{\beta}}^{ \mu} - {1 \over 4} F_{\mu \nu} F^{ \mu \nu} \eta_{\alpha \beta} \right )

Formulation_of_maxwell\\\'s_equations_in_special_relativity

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