en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-02T11:59:07Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 13:59, 2 May 2014
Line 1:		Line 1:
	In [[physics]], a '''ponderomotive force''' is a [[nonlinearity\|nonlinear]] [[force]] that a charged particle experiences in an inhomogeneous oscillating [[electromagnetic field\|electromagnetic]] field.		I am Oscar and I totally dig that name. Hiring is his profession. One of the issues she enjoys most is to do aerobics and now she is trying to make cash with it. Years in the past we moved to North Dakota.<br><br>My page; meal delivery service ([http://Bit.ly/1pABYYJ simply click the up coming document])

	~~The ponderomotive force '''F'''<sub>p</sub> is expressed by~~
	~~:[[ponderomotive force\|<math>\mathbf{F}_{\text{p}}=</math>]][[charged particle\|<math>-\frac{e^2}{4 m \omega^2}</math>]][[Del\|<math>\nabla</math>]][[amplitude\|<math>E^2</math>]]~~
	~~where ''e'' is the electrical [[electric charge\|charge]] of the particle, ''m'' is the mass, ''ω'' is the [[angular frequency]] of oscillation of the field,~~ and ~~''E'' is the [[amplitude]] of the electric field (at low enough amplitudes the magnetic field exerts very little force).~~

	~~This equation means that a charged particle in an inhomogeneous oscillating field not only oscillates at the frequency of ''ω'' but also drifts toward the weak field area. It is noteworthy~~ that ~~this is one rare case where the sign of the particle charge does not change the direction of the force, unlike the [[Lorentz force]].~~

	The mechanism of the ponderomotive force can be easily understood by considering the motion of the charge in an oscillating electric field. In the case of a homogeneous field, the charge returns to its initial position after one cycle of oscillation. In the case of an inhomogeneous field, the force exerted on the charge during the half-cycle it spends in the area with higher field amplitude points towards the weak field area. It is larger than the force exerted during the half-cycle spent in the area with a lower field amplitude, which points towards the strong field area. ~~Thus, averaged over a full cycle there~~ is ~~a net force that drives the charge toward the weak field area~~.

	~~==Derivation==~~

	~~The derivation~~ of the ~~ponderomotive force expression proceeds as follows.~~

	~~Consider a particle under the action of a non-uniform oscillating field. The equation of motion~~ is ~~given by:~~

	~~:<math>\ddot{x}=g(x)\cos(\omega t),</math>~~

	~~neglecting the effect of the associated oscillating magnetic field.~~

	If the length scale of variation of <math>g(x)</math> is large enough, then the particle trajectory can be divided into a slow time motion and a fast time motion:<ref>''Introduction to Plasma Theory'', second edition, by Nicholson, Dwight R., Wiley Publications (1983), ISBN 0-471-09045-X</ref>

	~~:<math>x=x_0+x_1</math>~~

	where <math>x_0</math> is the slow drift motion and <math>x_1</math> represents fast oscillations. Now, let us also assume that <math>x_1 \ll x_0</math>. Under this assumption, we can use Taylor expansion on the force equation about <math>x_0</math> to ~~get,~~

	~~:<math>\ddot{x_0}+\ddot{x_1}=\left[g(x_0)+x_1 g'(x_0)\right]\cos(\omega t)</math>~~

	~~:<math>\ddot{x_0} \ll \ddot{x_1}</math>,~~ and ~~because <math>x_1</math> is small, <math> g(x_0) \gg x_1 g'(x_0) </math>, so~~

	~~:<math>\ddot{x_1}=g(x_0)\cos(\omega t)</math>~~

	~~On the time scale on which <math>x_1</math> oscillates, <math>x_0</math>~~ is ~~essentially a constant. Thus, the above can be integrated~~ to ~~get,~~

	~~:<math>x_1=-\frac{g(x_0)}{\omega^2} \cos(\omega t)</math>~~

	~~Substituting this in the force equation and averaging over the <math>2\pi / \omega</math> timescale, we get,~~

	~~:<math>\ddot{x_0}=-\frac{g(x_0)g'(x_0)}{2 \omega^2}</math>~~

	~~:<math>\Rightarrow \ddot{x_0}=-\frac{1}{4 \omega^2}\left.\frac{d}{dx}\left[g(x)^2\right]\right\|_{x=x_0}</math>~~

	~~Thus, we have obtained an expression for the drift motion of a charged particle under the effect of a non-uniform oscillating field.~~

	~~==Time averaged Density==~~

	Instead of a single charged particle, there could be a gas of charged particles confined by the action of such a force. Such a gas of charged particles is called [[plasma (physics)\|plasma]]. The distribution function and density of the plasma will fluctuate at the applied oscillating frequency and to obtain an exact solution, we need to solve the [[Vlasov Equation]]. But, it is usually assumed that the time averaged density of the [[plasma (physics)\|plasma]] can be directly obtained from the expression for the force expression for the drift motion of individual charged particles:<ref>V. B. ~~Krapchev, ''Kinetic Theory of the Ponderomotive Effects~~ in ~~a Plasma'', Phys. Rev. Lett. 42, 497 (1979), ''http://prola.aps.org/abstract/PRL/v42/i8/p497_1''</ref>~~

	~~:<math>\bar{n}(x)=n_0 \exp \left[-\frac{e}{\kappa T} \Phi_{\text{P}} (x)\right]</math>~~

	~~where <math>\Phi_{\text{P}}</math> is the ponderomotive potential and is given by~~

	~~:<math>\Phi_{\text{P}} (x)=\frac{ m}{4 \omega^2} \left[g (x)\right]^2</math>~~

	~~==Generalized Ponderomotive Force==~~

	~~Instead of just an oscillating field, there could also be a permanent field present. In such a situation, the force equation of a charged particle becomes:~~

	~~:<math>\ddot{x}=h(x)+g(x)\cos(\omega t)</math>~~

	~~To solve~~ the ~~above equation,~~ we ~~can make a similar assumption as we did for the case when <math>h(x)=0</math>~~. ~~This gives a generalized expression for the drift motion of the particle:~~

	:<~~math~~>~~\ddot{x_0}=h(x_0)-\frac{g(x_0)g'(x_0)}{2 \omega^2}~~<~~/math~~>

	~~==Applications==~~

	~~The idea of a ponderomotive description of particles under the action of a time varying field has applications in areas like:~~

	~~# [[Quadrupole ion trap]]~~
	~~# [[Combined rf trap]]~~
	~~# [[Plasma acceleration]] of particles~~
	~~# [[Plasma propulsion engine]] especially the [[Electrodeless plasma thruster]]~~
	~~# [[High Harmonic Generation]]~~
	~~# [[Terahertz time-domain spectroscopy]] as a source of high energy THz radiation in laser-induced air plasmas~~

	~~The ponderomotive force also plays an important role in laser induced plasmas as a major density lowering factor.~~

	~~==References==~~
	~~;General~~
	* {{cite book \| author=Schmidt, George \| title=Physics of High Temperature Plasmas, second edition \| publisher =Academic Press \| year=1979 \| isbn=0-12-626660-3 \| page~~=47 }}~~
	;~~Citations~~
	~~{{reflist}}~~

	~~==Journals==~~

	*J. R. Cary and A. N. Kaufman, ''Ponderomotive effects in collisionless plasma: A Lie transform approach'', Phys. Fluids 24, 1238 (~~1981), ''http://dx.doi.org/10.1063/1.863527''~~
	*C. Grebogi and R. G. Littlejohn, ''Relativistic ponderomotive Hamiltonian'', Phys. Fluids 27, 1996 (1984), ''http://dx.doi.org/10.1063/1.864855''
	*G. J. Morales and Y. C. Lee, ''Ponderomotive-Force Effects in a Nonuniform Plasma'', Phys. Rev. Lett. 33, 1016 (1974), ''http://prola.aps.org/abstract/PRL/v33/i17/p1016_1''
	*B. M. Lamb and G. J. Morales, ''Ponderomotive effects in nonneutral plasmas'', Phys. Fluids 26, 3488 (1983), ''http://dx.doi.org/10.1063/1.864132''
	* K. Shah and H. Ramachandran, ''Analytic, nonlinearly exact solutions for an rf confined plasma'', Phys. Plasmas 15, 062303 (2008), ''http://~~link~~.~~aip.org/link/?PHPAEN/15/062303~~/~~1''~~
	* P. H. Bucksbaum, R. R. Freeman, M. Bashkansky, and T. J. McIlrath, ''Role of the ~~ponderomotive potential in above-threshold ionization'', Jour. Opt. Soc. B 4, 760 (1987~~)~~, ''http://www.opticsinfobase.org/josab/abstract.cfm?uri=josab-4-5-760''~~

	~~[[Category:Electrodynamics]]~~
	~~[[Category:Force]]~~

en>ChrisGualtieri: General Fixes + MOS + DMY changes using AWB

2013-06-30T04:40:29Z

General Fixes + MOS + DMY changes using AWB

Show changes

en>Jbergquist: /* Liquid/vapor equilibrium thermodynamic data */ corrected entry error row 23 col 9

2012-07-30T05:22:36Z

Liquid/vapor equilibrium thermodynamic data: corrected entry error row 23 col 9

Show changes

Carbon dioxide data - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

en>ChrisGualtieri: General Fixes + MOS + DMY changes using AWB

en>Jbergquist: /* Liquid/vapor equilibrium thermodynamic data */ corrected entry error row 23 col 9