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.

Details

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The electromagnetic tensor $\backslash ,\; F\_\{ab\}$ is commonly written as a matrix:

$$

F_{ab} = \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. When using natural units, the speed of light is taken to equal 1.

From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties (Mathematical note: In this article, the abstract index notation will be used.):

• antisymmetry: $F^\{ab\}\; \backslash ,\; =\; -\; F^\{ba\}$ (hence the name bivector).
• zero trace.
• six independent components.

More formally, the electromagnetic tensor may be written in terms of the four-potential $\backslash ,\; A^a$

$$

F_{ a b } \equiv \frac{ \partial A_b }{ \partial x^a } - \frac{ \partial A_a }{ \partial x^b } \equiv \partial_a A_b - \partial_b A_a \equiv A_{ b , a } - A_{ a , b }

where $A^a\; =\; (\; \backslash phi\; ,\; \backslash vec\; A\; c\; )$ and $A\_a\; \backslash ,\; =\; \backslash eta\_\{\; a\; b\; \}\; A^b$ ($\backslash ,\; \backslash eta$ is the Minkowski metric).

Derivation

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

$\backslash partial\_a\; =\; \backslash left(\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; \backslash right)\; =\; \backslash left(\backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\},\; \backslash nabla\; \backslash right)\; \backslash ,$

and

$A\_a\; =\; \backslash left(\backslash frac\{\backslash Phi\}\{c\},\; -A\_x,\; -A\_y,\; -A\_z\; \backslash right)\; \backslash ,$

where

$A\; \backslash ,$ is the vector potential

$\backslash Phi\; \backslash ,$ is the scalar potential and

$c\; \backslash ,$ is the speed of light

Electric and magnetic fields are derived from the vector potentials and the scalar potential with two formulas:

$\backslash vec\{E\}\; =\; -\backslash frac\{\backslash partial\; \backslash vec\{A\}\}\{\backslash partial\; t\}\; -\; \backslash nabla\; \backslash Phi\; \backslash ,$

$\backslash vec\{B\}\; =\; \backslash nabla\; \backslash times\; \backslash vec\{A\}\; \backslash ,$

As an example, the x components are just

$E\_x\; =\; -\backslash frac\{\backslash partial\; A\_x\}\{\backslash partial\; t\}\; -\; \backslash frac\{\backslash partial\; \backslash Phi\}\{\backslash partial\; x\}\; \backslash ,$

$B\_x\; =\; \backslash frac\{\backslash partial\; A\_z\}\{\backslash partial\; y\}\; -\; \backslash frac\{\backslash partial\; A\_y\}\{\backslash partial\; z\}\; \backslash ,$

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

$E\_x\; =\; -c\; \backslash left(\backslash partial\_1\; A\_0\; -\; \backslash partial\_0\; A\_1\; \backslash right)\; \backslash ,$

$B\_x\; =\; \backslash partial\_3\; A\_2\; -\; \backslash partial\_2\; A\_3\; \backslash ,$

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

$F\_\{a\; b\}\; =\; \backslash partial\_a\; A\_b\; -\; \backslash partial\_b\; A\_a\; \backslash ,$

Significance of the Field Tensor

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

$\backslash nabla\; \backslash cdot\; \backslash textbf\{\; E\; \}\; =\; \backslash frac\{\backslash rho\}\{\backslash epsilon\_0\}$

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

$$

\nabla \times \textbf{ B } - \frac{1}{c^2} \frac{ \partial \textbf{ E }}{\partial t} = \mu_0 \textbf{ 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 four pi times the current density.

These two equations for electricity reduce to

$\backslash partial\_a\; F^\{ab\}\; =\; \backslash mu\_0\; J^b$

where

$J^a\; =\; (\; \backslash rho\; ,\; J\; )\; \backslash ,$ is the 4-current.

The same holds for magnetism. If we take the magnetostatic's equation

$$

\nabla \cdot \textbf{ B } = 0

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

$$

\frac{ \partial \textbf{ B }}{ \partial t } + \nabla \times \textbf{ 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

$F\_\{\; a\; b\; ,\; c\; \}\; +\; F\_\{\; b\; c\; ,\; a\; \}\; +\; F\_\{\; c\; a\; ,\; b\; \}\; =\; 0.\; \backslash ,$

The field tensor and relativity

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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:

$F\_\{*\}\; \backslash ,\; =\; 0$ and $F^\{ab\}\{\}\_\{,b\}\; \backslash ,\; =\; \backslash mu\_0\; J^a$

where the comma indicates a partial derivative. The second equation implies conservation of charge:

$J^a\{\}\_\{,a\}\; \backslash ,\; =\; 0$

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

$F\_\{*\}\; \backslash ,\; =\; 0$ and $F^\{ab\}\{\}\_\{;b\}\; \backslash ,\; =\; \backslash mu\_0\; J^a$

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):

$J^a\{\}\_\{;a\}\; \backslash ,\; =\; 0$

Role in Quantum Electrodynamics and Field Theory

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The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity from $L\; =\; -\; \backslash frac\{1\}\{4\backslash pi\}F^\{\backslash mu\backslash nu\}F\_\{\backslash mu\backslash nu\},$ to incorporate the creation and annihilation of photons (and electrons).

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.

See also

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• Application of tensor theory in physics
• Classification of electromagnetic fields

References

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ElectromagnetismRelativityTensorsTensors in general relativity

Elektromagnetischer Feldstärketensor | טנזור השדה האלקטרומגנטי