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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. This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ...
Electromagnetism is the physics of the electromagnetic field; a field encompassing all of space 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. ...
Hermann Minkowski. ...
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. ...
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- where
- E is the electric field,
- B the magnetic field, and
- c the speed of light.
It has been suggested that optical field be merged into this article or section. ...
Current (I) flowing through a wire produces a magnetic field () around the wire. ...
A line showing the speed of light on a scale model of Earth and the Moon 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. It is the speed of all electromagnetic radiation...
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. ...
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...
A triangular number is a natural number such that the shape of an equilateral triangle can be formed by that number of points. ...
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. ...
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The product of the tensor 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. ...
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where is the completely antisymmetric unit tensor 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. ...
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More formally, the electromagnetic tensor may be written in terms of the 4-vector potential : Electromagnetic potential is . ...
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Where the 4-vector potential is: -
- and it's covariant form is found by multiplying by the Minkowski metric :
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: -
and the 4-vector potential: Electromagnetic potential is . ...
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where -
- is the vector potential and are its components
- is the scalar potential and
- 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. ...
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As an example, the x components are just -
Using the definitions we began with, we can re-write these two equations to look like: -
Evaluating all the components results in a second-rank, antisymmetric and covariant tensor: -
Relation to classical electromagnetism Classical electromagnetism and Maxwell's equations can be derived from the action defined: In electromagnetism, Maxwells equations are a set of equations, developed in the latter half of the nineteenth century by James Clerk Maxwell. ...
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where -
- 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. ...
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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. ...
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We can then plug this into the Euler-Lagrange equation of motion for a field: The Euler-Lagrange Equation is the major formula of the Calculus of variations. ...
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The second term is zero, because the lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes: -
That term in the parenthesis is just the field tensor, so this finally simplifies to -
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That equation is just another way of writing the two homogeneous Maxwell's equations as long as you make the subsitutions: -
where and 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 -
which tells us that the divergence of the electric field vector is equal to the charge density, and the electrodynamic equation -
that is the change of the electric field with respect to time, minus the curl of the magnetic field vector, is equal to negative four pi times the current density. In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ...
These two equations for electricity reduce to
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where -
- 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. ...
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which tells us that there are no "true" magnetic charges, and the magnetodynamics equation -
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 In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ...
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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. ...
The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...
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. ...
In electromagnetism, Maxwells equations are a set of equations, developed in the latter half of the nineteenth century by James Clerk Maxwell. ...
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- and
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. ...
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In general relativity, these laws can be generalised in (what many physicists consider to be) an appealing way: -
- and
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. ...
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Role in Quantum Electrodynamics and Field Theory The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity from 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 electromagnetism. ...
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 application of quantum mechanics to 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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