FACTOID # 3: South Carolina has the highest rate of violent crimes and aggravated assaults per capita among US states.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW

SEARCH ALL

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Electromagnetic tensor
Electromagnetism
Electricity · Magnetism
This box: view  talk  edit

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

The electromagnetic tensor Fαβ is commonly written as a matrix: Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...

$F_{alphabeta} = begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c E_x/c & 0 & B_z & -B_y E_y/c & -B_z & 0 & B_x E_z/c & B_y & -B_x & 0 end{bmatrix}$

or

$F^{alphabeta} = begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c -E_x/c & 0 & B_z & -B_y -E_y/c & -B_z & 0 & B_x -E_z/c & B_y & -B_x & 0 end{bmatrix}$
where
E is the electric field,
B the magnetic field, and
c the speed of light.

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... For the indie-pop band, see The Magnetic Fields. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ...

### Properties

From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties:

• antisymmetry: $F^{alphabeta} , = - F^{betaalpha}$ (hence the name bivector).
• six independent components.

If one forms an inner product of the field strength tensor a Lorentz invariant is formed: In set theory, the adjective antisymmetric usually refers to an antisymmetric relation. ... A bivector is an element of the antisymmetric tensor product of a tangent space with itself. ... Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ...

$F_{alphabeta} F^{alphabeta} = 2 left( B^2 - frac{E^2}{c^2} right) = mathrm{invariant}$

The product of the tensor $F^{alphabeta} ,$ with its dual tensor gives the pseudoscalar invariant: In mathematics, a pseudoscalar in a geometric algebra is the highest-grade basis element of the algebra. ...

$epsilon_{alphabetagammadelta}F^{alphabeta} F^{gammadelta} = frac{2}{c} left( vec B cdot vec E right) = mathrm{invariant} ,$

where $epsilon_{alphabetagammadelta} ,$ is the completely antisymmetric unit pseudotensor of the fourth rank or Levi-Civita symbol. Notice that The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

$det left( F right) = frac{1}{c^2} left( vec B cdot vec E right) ^{2}$

More formally, the electromagnetic tensor may be written in terms of the 4-vector potential $A^{alpha} ,$: Electromagnetic potential is . ...

$F_{ alphabeta } stackrel{mathrm{def}}{=} frac{ partial A_{beta} }{ partial x^{alpha} } - frac{ partial A_{alpha} }{ partial x^{beta} } stackrel{mathrm{def}}{=} partial_{alpha} A_{beta} - partial_{beta} A_{alpha}$

Where the 4-vector potential is:

$A^{alpha} = left( frac{phi}{c} , vec A right)$ and its covariant form is found by multiplying by the Minkowski metric $eta ,$:
$A_{alpha} , = eta_{alphabeta} A^{beta} = left( -frac{phi}{c}, vec A right)$

In category theory, see covariant functor. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

## Derivation of tensor

To derive all the elements in the electromagnetic tensor we need to define the derivative operator:

$partial_{alpha} = left(frac{1}{c} frac{partial}{partial t}, frac{partial}{partial x}, frac{partial}{partial y}, frac{partial}{partial z} right) = left(frac{1}{c} frac{partial}{partial t}, vec{nabla} right) ,$

and the 4-vector potential: Electromagnetic potential is . ...

$A_{alpha} = left(-frac{phi}{c}, A_x, A_y, A_z right) ,$

where

$vec A ,$ is the vector potential and $left(A_x, A_y, A_z right)$ are its components
$phi ,$ is the scalar potential and
$c ,$ is the speed of light.

Electric and magnetic fields are derived from the vector potentials and the scalar potential with two formulas: In vector calculus, a vector potential is a vector field whose curl is a given vector field. ... It has been suggested that this article or section be merged with Potential. ...

$vec{E} = -frac{partial vec{A}}{partial t} - vec{nabla} phi ,$
$vec{B} = vec{nabla} times vec{A} ,$

As an example, the x components are just

$E_x = -frac{partial A_x}{partial t} - frac{partial phi}{partial x} ,$
$B_x = frac{partial A_z}{partial y} - frac{partial A_y}{partial z} ,$

Using the definitions we began with, we can re-write these two equations to look like:

$E_x = -c left(partial_0 A_1 - partial_1 A_0 right) ,$
$B_x = partial_2 A_3 - partial_3 A_2 ,$

Evaluating all the components results in a second-rank, antisymmetric and covariant tensor:

$F_{alphabeta} = partial_{alpha} A_{beta} - partial_{beta} A_{alpha} ,$

Thus, for example,

$F_{12}=partial_1A_2-partial_2A_1=partial_xA_y-partial_yA_x=B_z,$

and

$F_{13}=partial_1A_3-partial_3A_1=partial_xA_z-partial_zA_x=-B_y.$

Compare with the matrix above.

## Relation to classical electromagnetism

Classical electromagnetism and Maxwell's equations can be derived from the action defined: For thermodynamic relations, see Maxwell relations. ...

$mathcal{S} = int left( -begin{matrix} frac{1}{4 mu_0} end{matrix} F_{munu} F^{munu} right) mathrm{d}^4 x ,$

where

$mathrm{d}^4 x ;$   is over space and time.

This means the Lagrangian is A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

 $mathcal{L} ,$ $= -begin{matrix} frac{1}{4mu_0} end{matrix} F_{munu} F^{munu} ,$ $= -begin{matrix} frac{1}{4mu_0} end{matrix} left( partial_mu A_nu - partial_nu A_mu right) left( partial^mu A^nu - partial^nu A^mu right). ,$ $= -begin{matrix} frac{1}{4mu_0} end{matrix} left( partial_mu A_nu partial^mu A^nu - partial_nu A_mu partial^mu A^nu - partial_mu A_nu partial^nu A^mu + partial_nu A_mu partial^nu A^mu right).$

The far left and far right term are the same, because μ and ν are just dummy variables after all. The two middle terms are also the same, so the Lagrangian is In computer programming, a free variable is a variable referred to in a function that is not a local variable or an argument of that function. ...

 $mathcal{L} ,$ $= -begin{matrix} frac{1}{2mu_0} end{matrix} left( partial_mu A_nu partial^mu A^nu - partial_nu A_mu partial^mu A^nu right).$

We can then plug this into the Euler-Lagrange equation of motion for a field: The Euler-Lagrange equation, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s, is the major formula of the calculus of variations. ...

$partial_nu left( frac{partial mathcal{L}}{partial ( partial_nu A_mu )} right) - frac{partial mathcal{L}}{partial A_mu} = 0 . ,$

The second term is zero, because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:

$partial_nu left( partial^mu A^nu - partial^nu A^mu right) = 0. ,$

That term in the parenthesis is just the field tensor, so this finally simplifies to

 $partial_nu F^{mu nu} = 0. ,$

That equation is just another way of writing the two homogeneous Maxwell's equations as long as you make the substitutions:

$~E^i /c = -F^{0 i} ,$
$epsilon^{ijk} B^k = -F^{ij} ,$

where $i ,$ and $j ,$ take on the values of 1, 2, and 3.

## Significance of the Field Tensor

Hidden beneath the surface of this complex mathematical equation is an ingenious unification of Maxwell's equations for electromagnetism. Consider the electrostatic equation

$vec{nabla} cdot vec{E} = frac{rho}{epsilon_0}$

which tells us that the divergence of the electric field vector is equal to the charge density, and the electrodynamic equation

$vec{nabla} times vec{B} - frac{1}{c^2} frac{ partial vec{E}}{partial t} = mu_0 vec{J}$

that is the change of the electric field with respect to time, minus the curl of the magnetic field vector, is equal to negative 4π times the current density. For other uses, see Curl (disambiguation). ...

These two equations for electricity reduce to

$partial_{beta} F^{alphabeta} = mu_0 J^{alpha} ,$

where

$J^{alpha} = ( c , rho , vec{J} ) ,$ is the 4-current.

The same holds for magnetism. If we take the magnetostatic equation In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ...

$vec{nabla} cdot vec{B} = 0$

which tells us that there are no "true" magnetic charges, and the magnetodynamics equation

$frac{ partial vec{B}}{ partial t } + vec{nabla} times vec{E} = 0$

which tells us the change of the magnetic field with respect to time plus the curl of the Electric field is equal to zero (or, alternatively, the curl of the electric field is equal to the negative change of the magnetic field with respect to time). With the electromagnetic tensor, the equations for magnetism reduce to For other uses, see Curl (disambiguation). ...

$F_{ alpha beta , gamma } + F_{ beta gamma , alpha } + F_{ gamma alpha , beta } = 0. ,$

## The field tensor and relativity

The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law, this general property of (non-gravitational) physical laws being recognised after the advent special relativity. This theory stipulated that all the (non-gravitational) laws of physics should take the same form in all coordinate systems - this led to the introduction of tensors. The tensor formalism also leads to a mathematically elegant presentation of physical laws. For example, Maxwell's equations of electromagnetism may be written using the field tensor as: A tensor is a generalization of the concepts of vectors and matrices. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... For thermodynamic relations, see Maxwell relations. ...

$F_{[alphabeta,gamma]} , = 0$ and $F^{alphabeta}{}_{,beta} , = mu_0 J^{alpha}$

where the comma indicates a partial derivative. The second equation implies conservation of charge: In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

$J^alpha{}_{,alpha} , = 0$

In general relativity, these laws can be generalised in (what many physicists consider to be) an appealing way:

$F_{[alphabeta;gamma]} , = 0$ and $F^{alphabeta}{}_{;beta} , = mu_0 J^{alpha}$

where the semi-colon represents a covariant derivative, as opposed to a partial derivative. The elegance of these equations stems from the simple replacing of partial with covariant derivatives, a practice sometimes referred to in the parlance of GR as 'replacing partial with covariant derivatives'. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime): In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

$J^alpha{}_{;alpha} , = 0$

## Role in Quantum Electrodynamics and Field Theory

The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity from $mathcal{L}=barpsi(ihbar c , gamma^alpha D_alpha - mc^2)psi -frac{1}{4 mu_0}F_{alphabeta}F^{alphabeta},$ to incorporate the creation and annihilation of photons (and electrons). A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

In quantum field theory, it is used for the template of the gauge field strength tensor. That is used in addition to the local interaction Lagrangian, nearly identical to its role in QED. Quantum field theory (QFT) is the quantum theory of fields. ...

Tensors are usede in Solid Mechanics ; if stress and strain are 3x3 matrixes , then Hooks Law which connects them with a constant has to be a tensor. ... In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. ...

## References

• Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0-19-514665-4.
• Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. Perseus Publishing. ISBN 0-201-50397-2.

Results from FactBites:

 - electromagnetic processes in dispersive media: a treatment based on the dielectric tensor by d. b. melrose, isbn ... (1266 words) Electromagnetic Processes in Dispersive Media: A Treatment Based on the Dielectric Tensor by D. Melrose, ISBN 0521410258 This text presents a systematic discussion of electromagnetic waves and radiation processes in a wide variety of media. The treatment, taken from the field of plasma physics, is based on the dielectric tensor; the authors unify approaches used in plasma physics and astrophysics on the one hand and in optics on the other.
 Tensors (848 words) Tensors are used in various parts of physics, both as abstract constructs in mathematical physics and for describing relations between quantities represented by matrices. Diffusion tensor imaging (DTI) is a method for imaging the fibrous structure of organs in the body, such as the brain, using the diffusivity of water. The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor or Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism.
More results at FactBites »

Share your thoughts, questions and commentary here