Electromagnetic tensor

The electromagnetic tensor , or tensor of Maxwell is the name of the mathematical object (a Tenseur) describing the structure of the electromagnetic Champ in a given point.

Definition

This tensor is defined within the framework of the mathematical formalism of the restricted Relativité, where with three space dimensions is assistant a temporal dimension. The objects vectorial S thus have four components, one thus speaks about Quadrivecteur. The electromagnetic tensor can be seen like a matrix 4× 4, of which elements are determined by a Quadrivecteur called Potentiel vector, usually noted has . The tensor of Maxwell, usually noted F is given by the formula

$F\_\; \{ab\}\; =\; \backslash \; partial\_a\; A\_b\; -\; \backslash \; partial\_b\; A\_a$.

This tensor is antisymmetric and of null trace.

Expression of the components

The electromagnetic tensor makes it possible to reconsider the Force of Lorentz being exerted on a particle charged. This force, F has as an expression

$\{\backslash \; mathbf\; \{F\}\}\; =\; Q\; \{\backslash \; mathbf\; \{E\}\}\; +\; Q\; \{\backslash \; mathbf\; \{v\}\}\; \backslash \; wedge\; \{\backslash \; mathbf\; \{B\}\}$.

In restricted relativity, its expression becomes

$f^a\; =\; Q\; F^a\; \{\}\; \_b\; u^b$,

where U is the Quadrivitesse of the particle considered. This makes it possible to reconstitute the components of the tensor of Maxwell in a system of Cartesian Coordonnées:

$F^a\; \{\}\; \_b\; =\; \backslash \; left\; (\backslash \; begin\; \{array\}\; \{rrrr\}$

0 & \ frac {1} {c^2} E^x & \ frac {1} {c^2} E^y & \ frac {1} {c^2} E^z \ \ E^x & 0 & B^z & - B^y \ \ E^y & - B^z & 0 & B^x \ \ E^z & B^y & - B^x & 0 \ end {array} \ right) . The expression of the components F depends on the Convention of signature of metric the used. On the assumption that this one is of the type (+---), there is

$F\_\; \{ab\}\; ^\; \{(+---)\}\; =\; \backslash \; left\; (\backslash \; begin\; \{array\}\; \{rrrr\}$

0 & E^x & E^y & E^z \ \ - E^x & 0 & - B^z & B^y \ \ - E^y & B^z & 0 & - B^x \ \ - E^z & - B^y & B^x & 0 \ end {array} \ right) . In the opposite case, with convention (- +++), there is

$F\_\; \{ab\}\; ^\; \{(-\; +++)\}\; =\; \backslash \; left\; (\backslash \; begin\; \{array\}\; \{rrrr\}$

0 & - E^x & - E^y & - E^z \ \ E^x & 0 & B^z & - B^y \ \ E^y & - B^z & 0 & B^x \ \ E^z & B^y & - B^x & 0 \ end {array} \ right) . The difference between these two notations disappears if one expresses the fields electric E and magnetic B according to the potential vector. The expression of F corresponds to

$F\_\; \{xy\}\; =\; \backslash \; partial\_x\; A\_y\; -\; \backslash \; partial\_y\; A\_x$.

In convention (- +++), that also corresponds to

$F^\; \{(-\; +++)\}\_\; \{xy\}\; =\; \backslash \; partial\_x\; A^y\; -\; \backslash \; partial\_y\; A^x$.

This expression corresponds to the component according to Z of rotational three-dimensional of has , which corresponds, according to the Maxwell's equations with B Z , in accordance with the expression of F in convention (- +++). In the same way, in convention (+---), there is

$F^\; \{(+---)\}\_\; \{xy\}\; =\; \backslash \; partial\_y\; A^x\; -\; \backslash \; partial\_x\; A^y$,

who corresponds according to what precedes with - B Z . In a similar way, there is

$F\_\; \{xt\}\; =\; \backslash \; partial\_x\; A\_t\; -\; \backslash \; partial\_t\; A\_x$.

In convention (- +++), this is written

$F^\; \{(-\; +++)\}\_\; \{xt\}\; =\; -\; c^2\; \backslash \; partial\_x\; A^t\; -\; \backslash \; partial\_t\; A^x$,

and thus corresponds to the component of E according to X , if one compares the electric Potentiel V to C 2 has T , whereas in convention (+---), there is

$F^\; \{(+---)\}\_\; \{xt\}\; =\; c^2\; \backslash \; partial\_x\; A^t\; +\; \backslash \; partial\_t\; A^x$,

who corresponds well to - E X .

The components contravariantes are expressed in the same way:

$F^\; \{ab\}\; \{\}\; ^\; \{(+---)\}\; =\; \backslash \; left\; (\backslash \; begin\; \{array\}\; \{rrrr\}$

0 & - \ frac {1} {c^2} E^x & - \ frac {1} {c^2} E^y & - \ frac {1} {c^2} E^z \ \ \ frac {1} {c^2} E^x & 0 & - B^z & B^y \ \ \ frac {1} {c^2} E^y & B^z & 0 & - B^x \ \ \ frac {1} {c^2} E^z & - B^y & B^x & 0 \ end {array} \ right) , and

$F^\; \{ab\}\; \{\}\; ^\; \{(-\; +++)\}\; =\; \backslash \; left\; (\backslash \; begin\; \{array\}\; \{rrrr\}$

0 & \ frac {1} {c^2} E^x & \ frac {1} {c^2} E^y & \ frac {1} {c^2} E^z \ \ - \ frac {1} {c^2} E^x & 0 & B^z & - B^y \ \ - \ frac {1} {c^2} E^y & - B^z & 0 & B^x \ \ - \ frac {1} {c^2} E^z & B^y & - B^x & 0 \ end {array} \ right) .

Dual tensor

The electromagnetic tensor being antisymmetric, it is about a Bivecteur. It is possible to deduce its dual bivector from it, F

• , by the formula

  $F^*\_\; \{ab\}\; =\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; epsilon\_\; \{abcd\}\; F^\; \{Cd\}$,

  where ε is the Tenseur of Levi-Civita, which gives

  $F\_\; \{ab\}\; ^\; \{*\; \backslash ;\; (+---)\}\; =\; \backslash \; left\; (\backslash \; begin\; \{array\}\; \{rrrr\}$

  0 & C B^x & C B^y & C B^z \ \ - C B^x & 0 & \ frac {1} {C} E^z & - \ frac {1} {C} E^y \ \ - C B^y & - \ frac {1} {C} E^z & 0 & \ frac {1} {C} E^x \ \ - C B^z & \ frac {1} {C} E^y & - \ frac {1} {C} E^x & 0 \ end {array} \ right) , and

  $F\_\; \{ab\}\; ^\; \{*\; \backslash ;\; (-\; +++)\}\; =\; \backslash \; left\; (\backslash \; begin\; \{array\}\; \{rrrr\}$

  0 & - C B^x & - C B^y & - C B^z \ \ C B^x & 0 & - \ frac {1} {C} E^z & \ frac {1} {C} E^y \ \ C B^y & \ frac {1} {C} E^z & 0 & - \ frac {1} {C} E^x \ \ C B^z & - \ frac {1} {C} E^y & \ frac {1} {C} E^x & 0 \ end {array} \ right) .

  In both cases, the operation of dualisation makes it possible to transform the electric field E into C B and the magnetic field B in - E / C .

  Mathematical aspects

  Mathematically, the tensor of Maxwell can be seen as the Dérivée external of has , which one can note in the form compacts

  $F\; =\; \{\backslash \; rm\; D\}\; A$.

  So the tensor of Maxwell can be defined by another potential vector, A' , defined by

  $A\text{'}\_a\; =\; A\_a\; +\; \backslash \; partial\_a\; \backslash \; phi$,

  or, more simply,

  $A\text{'}\; =\; has\; +\; \{\backslash \; rm\; D\}\; \backslash \; phi$,

  because the derivative external second of a quantity is null by definition. This property, the fact that the tensor of Maxwell is defined in a transformation close to the potential vector, is called Invariance of gauge.

  Maxwell's equations

  The two Maxwell's equations without source ($\backslash \; nabla\; \backslash \; cdot\; \{\backslash \; mathbf\; \{B\}\}\; =\; 0$ and $\backslash \; nabla\; \backslash \; wedge\; \{\backslash \; mathbf\; \{E\}\}\; =\; -\; \backslash \; partial\; \{\backslash \; mathbf\; \{B\}\}/\backslash \; partial\; t$) can be combined in only one very simple equation, namely

  $\{\backslash \; rm\; D\}\; F\; =\; 0$,

  who rises itself owing to the fact that F is already an external derivative, that the derivative external of an external derivative is identically null.

  The two equations implying the presence of loads, $\{\backslash \; rm\; div\}\; \{\backslash \; mathbf\; \{E\}\}\; =\; \backslash \; rho/\backslash \; epsilon\_0$ and $\{\backslash \; rm\; belch\}\; \{\backslash \; mathbf\; \{B\}\}\; =\; \backslash \; mu\_0\; \{\backslash \; mathbf\; \{J\}\}\; +\; (1/c^2)\; \backslash \; partial\; \{\backslash \; mathbf\; \{E\}\}/\backslash \; partial\; t$, can then be rewritten in the unified form

  $\{\}\; ^*\; \{\backslash \; rm\; D\}\; F^*\; =\; \backslash \; frac\; \{J\}\; \{\backslash \; epsilon\_0\}$,

  where J is the quadrivector of the electric current.

  See too

  • Electromagnetism

  • Maxwell's equations

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