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Encyclopedia > Electromagnetic tensor
Electromagnetism
Electricity · Magnetism
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The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism. The field tensor was first used after the 4-dimensional tensor formulation of special relativity introduced by Hermann Minkowski. The tensor allows some physical laws to be written in a very concise form. Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Image File history File links Solenoid. ... Electricity (from New Latin Ä“lectricus, amberlike) is a general term for a variety of phenomena resulting from the presence and flow of electric charge. ... For other senses of this word, see magnetism (disambiguation). ... Electrostatics (also known as static electricity) is the branch of physics that deals with the phenomena arising from what seem to be stationary electric charges. ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... This box:      Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each... 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. ... In physics, Gausss law gives the relation between the electric flux flowing out a closed surface and the charge enclosed in the surface. ... This article does not cite any references or sources. ... This article is about the electromagnetic phenomenon. ... Magnetostatics is the study of static magnetic fields. ... In physics, Ampères Circuital law, discovered by André-Marie Ampère, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ... This box:      Electric current is the flow (movement) of electric charge. ... For the indie-pop band, see The Magnetic Fields. ... Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... The Biot-Savart law is a physical law with applications in both electromagnetics and fluid dynamics. ... A bar magnet. ... Classical electrodynamics (or classical electromagnetism) is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell. ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ... Lorentz force. ... Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ... For magnetic induction, see Magnetic field. ... Faradays law of induction (more generally, the law of electromagnetic induction) states that the induced emf (electromotive force) in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. ... Displacement current is a quantity related to changing electric field. ... For thermodynamic relations, see Maxwell relations. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... This box:      Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. ... The Liénard-Wiechert potential describes the electromagnetic effect of a moving charge. ... In physics, the Maxwell stress tensor is the stress tensor of an electromagnetic field. ... As the circular plate moves down through a small region of constant magnetic field directed into the page, eddy currents are induced in the plate. ... A simple electric circuit made up of a voltage source and a resistor. ... Conduction is the movement of electrically charged particles through a transmission medium (electrical conductor). ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... An electric current i flowing around a circuit produces a magnetic field and hence a magnetic flux Φ through the circuit. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... A resonator is a device or part that vibrates (or oscillates) with waves. ... This box:      This page is about waveguides for electromagnetic wave propagation at microwave and radio wave frequencies. ... In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... In physics, the electromagnetic stress-energy tensor is the portion of the stress-energy tensor due to the electromagnetic field. ... 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. ... The electromagnetic four-potential is a four-vector defined in SI units (and gaussian units in parentheses) as in which φ is the electrical potential, and is the magnetic potential, a vector potential. ... André-Marie Ampère (January 20, 1775 – June 10, 1836), was a French physicist who is generally credited as one of the main discoverers of electromagnetism. ... Charles Augustin de Coulomb (born June 14, 1736, Angoulême, France - died August 23, 1806, Paris, France) was a French physicist. ... Michael Faraday, FRS (September 22, 1791 – August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed to the fields of electromagnetism and electrochemistry. ... Oliver Heaviside (May 18, 1850 – February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and... Joseph Henry Joseph Henry (December 17, 1797 – May 13, 1878) was a Scottish-American scientist who served as the first Secretary of the Smithsonian Institution. ... Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894) was the German physicist and mechanician for whom the hertz, an SI unit, is named. ... Hendrik Lorentz by Jan Veth Hendrik Antoon Lorentz (born July 18, 1853 in Arnhem, Netherlands; died February 4, 1928 in Haarlem, Netherlands) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... 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 a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Hermann Minkowski. ...

Contents

Details

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:

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. ...


See also

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. 

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