Siméon Denis Poisson - Revision history

Bbierbaum3 at 00:53, 6 December 2015

2015-12-06T00:53:47Z

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	===Early Life===		===Early Life===

	Poisson was born in Pithiviers, Loiret, France on June 21, 1781. The son of a soldier, he showed great promise in mathematics and science and started at Paris' École Polytechnique as first in his class. His focus was on mathematics, and at 18 was published in the esteemed journal ''Recueil des savants étrangers'' for his writings on finite difference equations.		Poisson was born in Pithiviers, Loiret, France on June 21, 1781. The son of a soldier, he showed great promise in mathematics and science and started at Paris' École Polytechnique as first in his class. His focus was on mathematics, and at 18 was published in the esteemed journal ''Recueil des savants étrangers'' for his writings on finite difference equations. He was a teaching assistant at the school and later a full professor. During his career he was published over 300 times.

	Poisson is most remembered for his work involving the application of mathematics to electricity, magnetism, mechanics, and other areas of physics. He is known for Poisson's equation, which is a partial differential equation that is useful in electrostatics, mechanical engineering and theoretical physics. For example, it can be used to describe the potential energy field caused by a given charge.		Poisson is most remembered for his work involving the application of mathematics to electricity, magnetism, mechanics, and other areas of physics. He is known for Poisson's equation, which is a partial differential equation that is useful in electrostatics, mechanical engineering and theoretical physics. For example, it can be used to describe the potential energy field caused by a given charge.
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	==Connectedness==		==Connectedness==
	~~#How is~~ this ~~topic connected~~ to ~~something that you are interested~~ in?		Poisson's accomplishments in this field highlight the ability for other schools of thought to assist each other in the collective further understanding of science. Poisson was traditionally a mathematician, but he decided to apply his knowledge to physics and was able to discover a new way of doing things.
	~~#How is it connected to your major?~~
	~~#Is there an interesting industrial application?~~

	~~==History==~~

	~~Put this idea in historical context~~. ~~Give the reader the Who, What, When, Where~~, and ~~Why~~.

	== See also ==		== See also ==

Bbierbaum3 at 00:17, 6 December 2015

2015-12-06T00:17:56Z

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	== See also ==		== See also ==

	[[~~http://physicsbook.gatech.edu/Electric_Potential\|~~Electric ~~potential~~]]		[[Electric Potential]]

	===Further reading===		===Further reading===

Bbierbaum3 at 00:15, 6 December 2015

2015-12-06T00:15:45Z

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	Siméon Poisson was a French mathematician best known for his work on definite integrals and electromagnetic theory.		Siméon Poisson was a French mathematician best known for his work on definite integrals and electromagnetic theory.
	[[File:Simeon Poisson.jpg\|thumb\|250px\|~~Simeon~~ Poisson]]		[[File:Simeon Poisson.jpg\|thumb\|250px\|Siméon Poisson]]

	==Personal Life==		==Personal Life==

Bbierbaum3 at 00:15, 6 December 2015

2015-12-06T00:15:25Z

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	== See also ==		== See also ==

	[[http://physicsbook.gatech.edu/Electric_Potential]Electric potential]		[[http://physicsbook.gatech.edu/Electric_Potential\|Electric potential]]

	===Further reading===		===Further reading===

Bbierbaum3 at 00:14, 6 December 2015

2015-12-06T00:14:40Z

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	== See also ==		== See also ==

	~~Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?~~		[[http://physicsbook.gatech.edu/Electric_Potential]Electric potential]

	===Further reading===		===Further reading===
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	===External links===		===External links===
	[~~http~~://www.~~scientificamerican~~.~~com~~/~~article~~/~~bring-science-home-reaction-time~~/]		[https://www.encyclopediaofmath.org/index.php/Poisson_equation]
			[http://planetmath.org/poissonsequation]

	==References==		==References==

	#https://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Simeon_Poisson.jpg/800px-Simeon_Poisson.jpg		#https://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Simeon_Poisson.jpg/800px-Simeon_Poisson.jpg
			#http://eqworld.ipmnet.ru/en/solutions/lpde/lpde302.pdf
			#http://www.britannica.com/biography/Simeon-Denis-Poisson
			#http://www.encyclopedia.com/topic/Simeon_Denis_Poisson.aspx

	~~[[Category:Which Category did you place this in?]]~~
	[[Category:Notable Scientists]]		[[Category:Notable Scientists]]

Bbierbaum3 at 23:27, 5 December 2015

2015-12-05T23:27:52Z

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	===Electrostatics===		===Electrostatics===

	~~'''~~Poisson's equation~~'''~~ for electrostatics is:		Assuming that the magnetic field is not changing with time, Poisson's equation for electrostatics is:

	:<math>{\nabla}^2 \varphi = -\frac{\rho_f}{\varepsilon}.</math>		:<math>{\nabla}^2 \varphi = -\frac{\rho_f}{\varepsilon}.</math>

	Solving Poisson's equation ~~for the potential requires knowing~~ the charge density distribution. If the charge density is zero, then Laplace's equation ~~results. If the charge density follows a Boltzmann distribution~~, ~~then the Poisson~~-~~Boltzmann equation results. The Poisson–Boltzmann equation plays a role in the development of the Debye–Hückel theory of dilute electrolyte solutions~~.		Solving for the potential using Poisson's equation necessitates knowledge of the charge density distribution. If the charge density comes out to be zero, then you get Laplace's equation, another differential equation named after Pierre-Simon Laplace.

	~~Using Green's Function, the~~ potential at distance ''r'' from a ~~central~~ point charge ''Q'' ~~(ie: the Fundamental Solution)~~ is:		The potential at a distance ''r'' from a point charge ''Q'' is:
	:<math>\varphi(r) = \dfrac {Q}{4 \pi \varepsilon r}.</math>		:<math>\varphi(r) = \dfrac {Q}{4 \pi \varepsilon r}.</math>
	~~(For historic reasons and unlike gravity's model above, the <math>4 \pi</math> factor appears here and not in Gauss's law.)~~

	The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the Coulomb gauge is used. In this more general context, computing ''φ'' is no longer sufficient to calculate '''E''', since '''E''' also depends on the magnetic vector potential '''A''', which must be independently computed.

	==Connectedness==		==Connectedness==

Bbierbaum3 at 23:21, 5 December 2015

2015-12-05T23:21:20Z

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	===A Mathematical Model===		===A Mathematical Model===

	Solving the Poisson equation lets one find the electric potential φ for a charge distribution ''<math>\rho_f</math>''. Poisson's equation is		Solving the Poisson equation lets one find the electric potential φ for a charge distribution ''<math>\rho_f</math>''. Poisson's equation is:

	:<math>\Delta\varphi=f</math>		:<math>\Delta\varphi=f</math>

	~~where <math>\Delta</math> is the Laplace operator,~~ and ''f'' and ''φ'' are real or complex-valued functions on a manifold. Usually, ''f'' is given and ''φ'' is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇<sup>2</sup> and so Poisson's equation is often written as		and can also be written as:

	:<math>\nabla^2 \varphi = f.</math>		:<math>\nabla^2 \varphi = f.</math>
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	===Electrostatics===		===Electrostatics===

	'''Poisson's equation''' for electrostatics~~, which~~ is:		'''Poisson's equation''' for electrostatics is:

	:<math>{\nabla}^2 \varphi = -\frac{\rho_f}{\varepsilon}.</math>		:<math>{\nabla}^2 \varphi = -\frac{\rho_f}{\varepsilon}.</math>

Bbierbaum3: /* Scientific Contributions */

2015-12-05T23:15:32Z

Scientific Contributions

← Older revision		Revision as of 19:15, 5 December 2015
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	===A Mathematical Model===		===A Mathematical Model===

	Poisson's equation is		Solving the Poisson equation lets one find the electric potential φ for a charge distribution ''<math>\rho_f</math>''. Poisson's equation is

	:<math>\Delta\varphi=f</math>		:<math>\Delta\varphi=f</math>

Bbierbaum3 at 23:07, 5 December 2015

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	:<math>\nabla^2 \varphi = f.</math>		:<math>\nabla^2 \varphi = f.</math>

	==Electrostatics==		===Electrostatics===

	'''Poisson's equation''' for electrostatics, which is:		'''Poisson's equation''' for electrostatics, which is:
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	(For historic reasons and unlike gravity's model above, the <math>4 \pi</math> factor appears here and not in Gauss's law.)		(For historic reasons and unlike gravity's model above, the <math>4 \pi</math> factor appears here and not in Gauss's law.)

	The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the [[Coulomb gauge]] is used. In this more general context, computing ''φ'' is no longer sufficient to calculate '''E''', since '''E''' also depends on the [[magnetic vector potential]] '''A''', which must be independently computed. See [[Mathematical descriptions of the electromagnetic field#Maxwell's equations in potential formulation\|Maxwell's equation in potential formulation]] for more on ''φ'' and '''A''' in Maxwell's equations and how Poisson's equation is obtained in this case.		The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the Coulomb gauge is used. In this more general context, computing ''φ'' is no longer sufficient to calculate '''E''', since '''E''' also depends on the magnetic vector potential '''A''', which must be independently computed.

	~~=== Potential of a Gaussian charge density ===~~
	~~Where ''Q'' is the total charge, the solution ''φ''(''r'') of Poisson's equation for this situation is given by~~

	~~:<math> \varphi(r) = { 1 \over 4 \pi \varepsilon } \frac{Q}{r}\,\mbox{erf}\left(\frac{r}{\sqrt{2}\sigma}\right)</math>~~

	~~where erf(''x'') is the error function~~.

	==Connectedness==		==Connectedness==

Bbierbaum3 at 23:03, 5 December 2015

2015-12-05T23:03:31Z

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	Poisson was born in Pithiviers, Loiret, France on June 21, 1781. The son of a soldier, he showed great promise in mathematics and science and started at Paris' École Polytechnique as first in his class. His focus was on mathematics, and at 18 was published in the esteemed journal ''Recueil des savants étrangers'' for his writings on finite difference equations.		Poisson was born in Pithiviers, Loiret, France on June 21, 1781. The son of a soldier, he showed great promise in mathematics and science and started at Paris' École Polytechnique as first in his class. His focus was on mathematics, and at 18 was published in the esteemed journal ''Recueil des savants étrangers'' for his writings on finite difference equations.

	~~===Life in Academia===~~

	Poisson is most remembered for his work involving the application of mathematics to electricity, magnetism, mechanics, and other areas of physics. He is known for Poisson's equation, which is a partial differential equation that is useful in electrostatics, mechanical engineering and theoretical physics. For example, it can be used to describe the potential energy field caused by a given charge.		Poisson is most remembered for his work involving the application of mathematics to electricity, magnetism, mechanics, and other areas of physics. He is known for Poisson's equation, which is a partial differential equation that is useful in electrostatics, mechanical engineering and theoretical physics. For example, it can be used to describe the potential energy field caused by a given charge.