Classification Of Electromagnetic Fields

Classification of electromagnetic fields - Wikipedia, the free encyclopedia
Classification of electromagnetic fields. From Wikipedia, the free encyclopedia ... so we immediately obtain an algebraic classification of electromagnetic fields. ...
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Electromagnetic field - Wikipedia, the free encyclopedia
The field propagates by electromagnetic radiation; in ... Classification of electromagnetic fields. Electric field. Electromagnetism. Electromagnetic tensor ...
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electromagnetic field: Definition from Answers.com
electromagnetic field n. The field of force associated with electric charge in motion, having both ... Classification of electromagnetic fields. Electric ...
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DOE Document - CLASSIFICATION OF ELECTROMAGNETIC AND GRAVITATIONAL FIELDS.
... to historical and current research (1948 to the present) from the Department of Energy (DOE) and predecessor ... CLASSIFICATION OF ELECTROMAGNETIC AND ...
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Electromagnetic Fields (EHC 137, 1992)
... known biological effects of electromagnetic fields in the frequency range 300 ... However, not all electromagnetic field effects can be explained in terms of the ...
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WHO | Electromagnetic fields and public health
This classification was based on pooled analyses of epidemiological studies ... static and low frequency electromagnetic fields, biological effects and health ...
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In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold.

It is used in the study of solutions of Maxwell's equations and has applications in Einstein's theory of general relativity, but the theorem is a purely mathematical one.

The classification theorem A (real) bivector field may be viewed, at any given event in a spacetime, as a skew-symmetric linear operator on a four-dimensional (real) vector space, r^a \, \rightarrow F^a{}_b r^b. Here, the vector space is the tangent space at the given event, and thus isomorphic as a (real) inner product space to E1,3. That is, it has the same notion of vector magnitude (mathematics) and angle (or inner product) as Minkowski spacetime.

In the remainder of this section (and in the next section), we'll assume our spacetime is Minkowski spacetime. This simplifies the mathematics (but tends to blur the distinction between the tangent space at an event and the underlying manifold). Fortunately, nothing will be lost by this apparently drastic specialization, for reasons we discuss as the end of the article.

In studying any linear operator, we confront the eigenvalue problem, that is, the problem of finding eigenvalues \lambda and eigenvectors r which satisfy the eigenvalue equation F^a{}_br^b \, =\lambda r^a The skew-symmetry of the operator we are interested in now implies that one of the following must hold:

r is a null vector (Minkowski space) belonging to a nonzero eigenvalue
r is a nonnull eigenvector belonging to the eigenvalue zero
r is a null eigenvector belonging to the eigenvalue zero

The linearly independent null eigenspaces are called the principal null directions of the bivector.

The classification theorem characterizes the possible principal null directions of a bivector. It states that one of the following must hold for any nonzero bivector:

one repeated principal null direction, in this case, the bivector is said to be null,
two distinct principal null directions, in this case, the bivector is said to be non-null.

Furthermore, for any non-null bivector, the two eigenvalues associated with the two distinct principal null directions have the same magnitude but opposite sign, \lambda=\pm \nu, so we have three subclasses of non-null bivectors: *spacelike: \nu \, =0 *timelike :\nu \neq 0 and \operatorname{rank} \, F = 2 *non-simple :\nu \neq 0 and \operatorname{rank} \, F = 4 where the rank refers to the rank (linear algebra) of the linear operator F. Every nonsimple bivector can be written as a sum of at most two simple ones.

Physical interpretation The algebraic classification of bivectors given above has an important application in relativistic physics: the electromagnetic field is represented by a skew-symmetric second rank tensor (the electromagnetic field tensor) so we immediately obtain an algebraic classification of electromagnetic fields.

Recall that for in a cartesian chart on Minkowski spacetime, the electromagnetic field tensor has components F_{ab} = \left( \begin{matrix}0 & B_z & -B_y & E_x/c \\-B_z & 0 & B_x & E_y/c \\B_y & -B_x & 0 & E_z/c \\-E_x/c & -E_y/c & -E_z/c & 0\end{matrix}\right) where E_x, E_y, E_z and B_x, B_y, B_z denote respectively the components of the electric and magnetic fields, as measured by an inertial observer (at rest in our coordinates). As usual in relativistic physics, we will find it convenient to work with geometrized unit system in which c=1. In the "tensor gymanastics" formalism of special relativity, the Minkowski metric \eta is used to raise and lower indices.

Invariants The fundamental invariants of the electromagnetic field are: P \equiv \frac{1}{2} F_{ab} \, F^{ab} = \| \vec{B} \|^2 - \| \vec{E} \|^2 = -\frac{1}{2}{}^* F_{ab} \, {}^* F^{ab} Q \equiv \frac{1}{4} F_{ab} \, {}^*F^{ab} = \vec{E} \cdot \vec{B}. (Fundamental means that every other invariant can be expressed in terms of these two.)

A null electromagnetic field is characterised by P = Q =0. In this case, the invariants reveal that the electric and magnetic fields are perpendicular and that they are of the same magnitude (in geometrised units). An example of a null field is a plane wave in Minkowski space.

A non-null field is characterised by P^2+Q^2 \neq \, 0. If P \neq 0 = Q, there exists an inertial frame for which either the electric or magnetic field vanishes. (These correspond respectively to magnetostatic and electrostatic fields.) If Q \neq 0, there exists an inertial frame in which electric and magnetic fields are proportional.

Curved Lorentzian manifolds So far we have discussed only flat spacetime, i.e. the Minkowski spacetime. Fortunately, according to the (strong) equivalence principle, if we simply replace "inertial frame" above with a frame fields in general relativity, everything works out exactly the same way on curved manifolds.

See also

References

See section 25.