This definition of polarization as a "dipole moment per unit volume" is widely adopted,
though in some cases it can bring to ambiguities and
paradoxes
.[1]
Relation among E, P, D
In this equation, P is the (negative of the) field induced in the material when the "fixed" charges, the dipoles, shift in response to the total underlying field E, whereas D is the field due to the remaining charges, known as "free" charges
[1]
.[2]
Polarization ambiguity
Another problem in the definition of is related to the arbitrary choice of the "unit volume", or more precisely to the system's scale
.[1]
For example, at microscopic scale a plasma can be regarded as a gas of free charges,
thus should be zero. On the contrary, at a macroscopic scale the same plasma
can be described as a continuous media, exhibiting a permittivity
and thus a net polarization .
so that the total current density that enters Maxwell's equations is given by
where Jf is the electric current density of free charges (also called the free current), the second term is the contribution from the magnetization, and the last term is related to the electric polarizationP.
Applications
The cross product has applications in different contexts,
e.g. it is used in computational geometry, physics and engineering.
A non-exhaustive list of examples is reported.
Angular momentum and torque
The angular momentum of
a particle about a given origin is defined as:
where is the position vector of the particle relative to the origin, is the linear momentum of the particle.
In the same way, the moment of a force
applied at point B around point A is given as:
In Mechanics the moment of a force is also called torque and written as
Since position , linear momentum
and force are all true vectors,
both the angular momentum
and the moment of a force
are pseudovectors or axial vectors.
Rigid body
The cross product frequently appears in the description of rigid motions.
Since velocity , force
and electric field are all true vectors,
the magnetic field is a pseudovector.
Generalizations
Skew-symmetric matrix
If the cross product is defined as a binary operation, it takes in input just 2 vectors.
If its output is not required to be a vector or a pseudovector but a matrix, then
it can be generalized in an arbitrary number of dimensions
[3][5][6]
.
where is formally defined from the rotation matrix
associated to body's frame: .
In three-dimensions holds:
In Quantum Mechanics the angular momentum is often represented as an anti-symmetric matrix or tensor. More precisely, it is the result of cross product involving position and linear momentum :
Since both and can have an arbitrary
number of components, that kind of cross product can be extended to any dimension,
holding the "physical" interpretation of the operation.
- Sistemare la citazione [1] del lavoro di Alù e Monticone (mancano i riferimenti alla pubblicazione)
[8]
- Nella sezione "Metamaterial cloaking", dopo la frase:
"Using transformation optics it is possible to design the optical parameters of a "cloak" so that it guides light around some region, rendering it invisible over a certain band of wavelengths."
aggiungere il riferimento all'articolo di Pendry e Smith, su "Controlling EM fields"
[9]
Poco dopo: "There are several theories of cloaking, giving rise to different types of invisibility."
aggiungere riferimenti a tesi, Alù/Engheta, Tachi.
[10][11][12][13]
-References
numerare la citazione di Ulf e Smith e sistemare dopo quella di Pendry su "Controlling EM fields"
[14]
Invisibility
Sezione "Pratical efforts"
"Engineers and scientists have performed various kinds of research to investigate the possibility of finding ways to create real optical invisibility (cloaks) for objects. Methods are typically based on implementing the theoretical techniques of transformation optics, which have given rise to several theories of cloaking."
Aggiungere un riferimento alla tesi, a Pendry,Smith, Galdi e Alù.
Sezione "External links":
aggiungere link alla Tachi (vedi pagina del "Cloaking Device")
Cloak of invisibility
Sezione "References"
Sezione "Further readings"
Aggiungere un riferimento alla tesi.
aggiungere link alla Tachi
((cite book
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|last="cognome"
|title="titolo"
|year="anno"
|publisher="casa_ed."
|location="città"
))
. 1982. ((cite book)): Missing or empty |title= (help); Text "Ballanti" ignored (help); Text "Federico" ignored (help); Text "Lato Side Editori" ignored (help); Text "Led Zeppelin" ignored (help); Text "Roma" ignored (help)
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Levi-Civita, T.; Amaldi, U. (1949). Lezioni di meccanica razionale (in italiano). Vol. 1. Bologna: Zanichelli editore.((cite book)): CS1 maint: unrecognized language (link)
Morando, A.P.; Leva, S. (1998). Note di teoria dei Campi Vettoriali (in italiano). Bologna: Esculapio.((cite book)): CS1 maint: unrecognized language (link)
A.T. de Hoop (2012). "Lorentz-covariant electromagnetic fields in (N + 1)-spacetime — An axiomatic approach to special relativity". Wave Motion. 49 (8). Elsevier: 737–744. doi:10.1016/j.wavemoti.2012.05.002.