Definitions

Polarization Magnetization

Definition P

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 .


[3]

[4] [5] [6] .

Magnetization

Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. [1] It is represented by a pseudovector M.

Definition

Where dp is the elementary electric dipole moment.

Those definitions of P and M as a "moments per unit volume" are widely adopted, though in some cases they can bring to ambiguities and paradoxes.[1]

Physics applications

To calculate the dipole moment m (A m2) using the formula: , we have that , thus , where:

[7]

Magnetization current

The magnetization M makes a contribution to the current density J, known as the magnetization current or bound (volumetric) current

[2]

and for the bound surface current:

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 polarization P.

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.

For two points P e Q on rigid body holds the law:

where is the point's position, is its velocity and is body's angular velocity.

Since position and velocity are true vectors, the angular velocity is a pseudovector or axial vector.

Lorentz force

See also: Lorentz force

The electromagnetic force exerted on a particle is:

where:

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] .

In Mechanics, for example, the angular velocity can be interpreted either as a pseudovector or as a anti-symmetric matrix or skew-symmetric tensor . In the latter case, the velocity law for a rigid body looks:

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.

See the "Alternative ways to compute the cross product" section for numerical details.

Cloaking device

- 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

Sezione "External links": aggiungere link al video di Alù su "The quest for invisibility"

aggiungere link alla Tachi (vedi pagina del "Cloaking Device")

Esempio per le note

Esempio 1: [15] [16]

Esempio 2: Paolo Maldini,[17] Fabio Cannavaro, Dino Zoff.[18]

See also

Notes

Bibliografia

((cite book| "nome" | "cognome" | "titolo" | "anno" | "casa_ed." | "città"))

oppure, con uno schema simile a quello di Bibtex:

((cite book |first="nome" |last="cognome" |title="titolo" |year="anno" |publisher="casa_ed." |location="città" ))


[4]

[19]

References

  1. ^ a b c d e C.A. Gonano; R.E. Zich; M. Mussetta (2015). "Definition for Polarization P and Magnetization M Fully Consistent with Maxwell's Equations" (PDF). Progress in Electromagnetics Research B. 64: 83–101. doi:10.2528/PIERB15100606.
  2. ^ a b A. Herczynski (2013). "Bound charges and currents" (PDF). American Journal of Physics. 81 (3): 202–205. doi:10.1119/1.4773441.
  3. ^ a b A.W. McDavid; C.D. McMullen (2006). "Generalizing Cross Products and Maxwell's Equations to Universal Extra Dimensions" (PDF). ((cite journal)): Cite journal requires |journal= (help)
  4. ^ a b WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537.
  5. ^ a b C.A. Gonano (2011). Estensione in N-D di prodotto vettore e rotore e loro applicazioni (PDF). Politecnico di Milano, Italy.
  6. ^ a b C.A. Gonano; R.E. Zich (2014). "Cross product in N Dimensions - the doublewedge product" (PDF). ((cite journal)): Cite journal requires |journal= (help)
  7. ^ https://www.kjmagnetics.com/glossary.asp
  8. ^ Monticone, F.; Alù, A. (2013). "Do Cloaked Objects Really Scatter Less?". Phys. Rev. X. 3 (4). American Physical Society. doi:10.1103/PhysRevX.3.041005. S2CID 118637398.
  9. ^ Pendry, J.B.; Schurig, D.; Smith, D.R. (2006). "Controlling electromagnetic fields" (PDF). Science. 312 (5781). American Association for the Advancement of Science: 1780–1782. doi:10.1126/science.1125907. PMID 16728597. S2CID 7967675.
  10. ^ Tachi, Susumu (2003). "Telexistence and retro-reflective projection technology (RPT)". Proceedings of the 5th Virtual Reality International Conference (VRIC2003) Pp. 69. Citeseer. CiteSeerX 10.1.1.97.221.
  11. ^ Inami, M.; Kawakami, N.; Susumu, T. (2003). "Optical camouflage using retro-reflective projection technology" (PDF). Proceedings of the 2nd IEEE/ACM International Symposium on Mixed and Augmented Reality. IEEE Computer Society: 348–349. doi:10.1109/ISMAR.2003.1240754. ISBN 0-7695-2006-5. S2CID 44776407.
  12. ^ Alù, A.; Engheta, N. (2008). "Plasmonic and metamaterial cloaking: physical mechanisms and potentials". Journal of Optics A: Pure and Applied Optics. 10 (9). IOP Publishing: 093002. doi:10.1088/1464-4258/10/9/093002.
  13. ^ Gonano, C.A. (2016). A perspective on metasurfaces, circuits, holograms and invisibility (PDF). Politecnico di Milano, Italy.
  14. ^ Leonhardt, Ulf; Smith, David R. (2008). "Focus on Cloaking and Transformation Optics". New Journal of Physics. 10 (11): 115019. Bibcode:2008NJPh...10k5019L. doi:10.1088/1367-2630/10/11/115019.
  15. ^ Tua fonte
  16. ^ Articolo del The New York Times
  17. ^ Calciatore attivo ma che ha rinunciato alla Nazionale.
  18. ^ Ha detenuto in passato il record di presenze in Nazionale.
  19. ^ T. Levi-Civita; U. Amaldi (1949). Lezioni di meccanica razionale (in Italian). Bologna: Zanichelli editore.

[1]

Voci correlate

Collegamenti esterni

Esempio 1: * http://www.google.com/

Esempio 2: * ((en)) [http://www.google.co.uk/ Il mio testo]



Category:Bilinear maps Category:Binary operations Category:Vector calculus Category:Analytic geometry


  1. ^ WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537.