formulasearchengine - User contributions [en]

Cationic polymerization

2014-12-09T21:44:07Z

92.224.253.221: /* Lewis acids/Friedel-Crafts catalysts */

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Symbol rate

2014-01-31T20:12:52Z

92.224.234.18: /* Line codes for baseband transmission */

In [[plasma physics]], an '''electromagnetic electron wave''' is a [[Waves in plasmas|wave]] in a [[plasma (physics)|plasma]] which has a [[magnetic field]] component and in which primarily the [[electron]]s oscillate.

In an unmagnetized plasma, an electromagnetic electron wave is simply a [[light]] wave modified by the plasma. In a magnetized plasma, there are two modes perpendicular to the field, the O and X modes, and two modes parallel to the field, the R and L waves.

==Cut-off frequency and critical density==
In an unmagnetized plasma in the high frequency or low density limit, i.e. for <math>\omega >> (4\pi n_ee^2/m_e)^{1/2}</math>
or
<math>n_e << m_e\omega^2\,/\,4\pi e^2</math>,
the wave speed is the [[speed of light]] in vacuum. As the density increases, the [[phase velocity]] increases and the [[group velocity]] decreases until the '''[[cut-off frequency]]''' where the light frequency is equal to the plasma frequency. This density is known as the '''critical density''' for the [[angular frequency]] ω of that wave and is given by <ref name=Chen>{{cite book|last=Chen|first=Francis|title=Introduction to Plasma Physics and Controlled Fusion, Volume 1|publisher=Plenum Publishing Corporation|year=1984|page=116|edition=2nd|isbn=0-306-41332-9}}</ref>

:<math>n_c = \frac{\varepsilon_o\,m_e}{e^2}\,\omega^2</math>.

If the critical density is exceeded, the plasma is called '''over-dense'''.

In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.

==O wave==
The '''O wave''' is the "ordinary" wave in the sense that its [[dispersion relation]] is the same as that in an unmagnetized plasma. It is [[Plane polarization|plane polarized]] with
'''E'''1 || '''B'''0.
It has a cut-off at the [[plasma frequency]].

==X wave==
The '''X wave''' is the "extraordinary" wave because it has a more complicated dispersion relation. It is partly transverse (with '''E'''1⊥'''B'''0)
and partly longitudinal. As the density is increased, the phase velocity rises from ''c'' until the cut-off at ωR is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency ωh. Then it can propagate again until the second cut-off at ωL. The cut-off frequencies are given by <ref name=Chen2>{{cite book|last=Chen|first=Francis|title=Introduction to Plasma Physics and Controlled Fusion, Volume 1|publisher=Plenum Publishing Corporation|year=1984|page=127|edition=2nd|isbn=0-306-41332-9}}</ref>
:<math>\omega_R = \frac{1}{2}\left[ \omega_c + (\omega_c^2+4\omega_p^2)^{1/2} \right]</math>
:<math>\omega_L = \frac{1}{2}\left[ -\omega_c + (\omega_c^2+4\omega_p^2)^{1/2} \right]</math>
where <math>\omega_c</math> is the [[electron cyclotron resonance]] frequency, and <math>\omega_p</math> is the electron [[plasma frequency]].

==R wave and L wave==
The '''R wave''' and the '''L wave''' are right-hand and left-hand circularly polarized, respectively. The R wave has a cut-off at ωR (hence the designation of this frequency) and a resonance at ωc. The L wave has a cut-off at ωL and no resonance. R waves at frequencies below ωc/2 are also known as '''whistler modes'''. <ref name=Chen3>{{cite book|last=Chen|first=Francis|title=Introduction to Plasma Physics and Controlled Fusion, Volume 1|publisher=Plenum Publishing Corporation|year=1984|page=131|edition=2nd|isbn=0-306-41332-9}}</ref>

==Dispersion relations==
The [[dispersion relation]] can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the [[index of refraction]] ''ck''/ω (squared).

{| class="wikitable"
|+ Summary of electromagnetic electron waves
|-
! conditions !! dispersion relation !! name
|-
| <math>\vec B_0=0</math> || <math>\omega^2=\omega_p^2+k^2c^2</math> || light wave
|-
| <math>\vec k\perp\vec B_0,\ \vec E_1\|\vec B_0</math> || <math>\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}</math> || O wave
|-
| <math>\vec k\perp\vec B_0,\ \vec E_1\perp\vec B_0</math> || <math>\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}\,
\frac{\omega^2-\omega_p^2}{\omega^2-\omega_h^2}</math> || X wave
|-
| <math>\vec k\|\vec B_0</math> ([[Polarization (waves)|right circ. pol.]])|| <math>\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1-(\omega_c/\omega)}</math> || R wave (whistler mode)
|-
| <math>\vec k\|\vec B_0</math> ([[Polarization (waves)|left circ. pol.]])|| <math>\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1+(\omega_c/\omega)}</math> || L wave
|}

==References==
{{Reflist|35em}}

==See also==
* [[Appleton-Hartree equation]]
* [[List of plasma (physics) articles]]

{{DEFAULTSORT:Electromagnetic Electron Wave}}
[[Category:Waves in plasmas]]

Pro-p group

2013-07-01T10:31:26Z

92.224.252.53:

'''Span''' is the distance between two intermediate supports for a structure, e.g. a [[beam (structure)|beam]] or a [[bridge]].
A span can be closed by a solid beam or by a rope. The first kind is used for bridges, the second one for [[electric power transmission|power line]]s, overhead telecommunication lines, some type of [[Antenna (radio)|antennas]] or for [[aerial tramway]]s.

[[Image:Bending.svg|frame|right|Side view of a [[Simply supported#Beams|simply supported]] beam (top) bending under an evenly distributed load (bottom).]]
The span is a significant factor in finding the strength and size of a beam as it determines the maximum [[bending moment]] and [[Deflection (engineering)|deflection]]. The maximum bending moment <math>M_{max}</math> and deflection <math>\delta_{max}</math>in the pictured beam is found using:<ref name='gere'>{{Cite book|title=Mechanics of Materials |edition=Eighth| isbn = 978-1-111-57773-5|last1=Gere|first1= James M.|last2=Goodno|first2=Barry J.|page=1086}}</ref>

:<math>M_{max} = \frac {q L^2} {8}</math>

:<math>\delta_{max} = \frac {5 M_{max} L^2} {48 E I} = \frac {5 q L^4} {384 E I}</math>

where

:<math>q</math> = Uniformly distributed load
:<math>L</math> = Length of the beam
:<math>E</math> = [[Modulus of elasticity]]
:<math>I</math> = [[Area moment of inertia]]

Note that the maximum bending moment and deflection occur midway between the two supports. From this it follows that if the span is doubled, the maximum moment (and with it the [[tensile stress|stress]]) will quadruple, and deflection will increase by a factor of sixteen.

For long-distance rope spans, used as power line, antenna or for aerial tramways, see [[list of spans]].

==References==
{{Reflist}}

{{DEFAULTSORT:Span (Architecture)}}
[[Category:Architectural elements]]
[[Category:Structural engineering]]

Uniformization (probability theory)

2012-06-24T16:49:29Z

92.224.240.158:

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