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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==
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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.
 
===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.
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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>


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>


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


'''Poisson's equation''' for electrostatics, which 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. 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.
=== Potential of a Gaussian charge density ===
If there is a static spherically symmetric [[Gaussian distribution|Gaussian]] charge density
:<math> \rho_f(r) = \frac{Q}{\sigma^3\sqrt{2\pi}^3}\,e^{-r^2/(2\sigma^2)},</math>
where ''Q'' is the total charge, then the solution ''φ''(''r'') of Poisson's equation,
:<math>{\nabla}^2 \varphi = - { \rho_f \over \varepsilon } </math>,
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]].
This solution can be checked explicitly by evaluating <math>{\nabla}^2 \varphi</math>. Note that, for ''r'' much greater than ''σ'', the erf function approaches unity and the potential φ (''r'') approaches the [[electrical potential|point charge]] potential
:<math> \varphi \approx { 1 \over 4 \pi \varepsilon } {Q \over r} </math>,
as one would expect. Furthermore the erf function approaches 1 extremely quickly as its argument increases; in practice for r > 3''σ'' the relative error is smaller than one part in a thousand.


==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 ==


Are there related topics or categories in this wiki resource for the curious reader to explore?  How does this topic fit into that context?
[[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]]

Latest revision as of 19:53, 5 December 2015

Created by Benjamin Bierbaum

Siméon Poisson was a French mathematician best known for his work on definite integrals and electromagnetic theory.

Siméon Poisson

Personal 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. 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.

Family Life

Poisson married Nancy de Bardi in 1817. Together they had four children.

Death and Legacy

Poisson's health was weak throughout his lifetime - he had several older siblings that died during childhood, and he was entrusted to a nurse during his early life. His health declined rapidly in 1840, and although extremely impaired, he continued to attend meetings of the French Academy of Sciences.

Poisson died on April 25, 1840. Attendees of his funeral included numerous French scientists, as well as the youngest son of King Louis Philippe I, who studied under Poisson.

Poisson was President of the French Academy of Sciences at the time of his death, and was also a member of the Royal Society of London. His name is inscribed on the Eiffel Tower in Paris alongside 71 other prominent French scientists, mathematicians, and engineers.

Scientific Contributions

A Mathematical Model

Solving the Poisson equation lets one find the electric potential φ for a charge distribution [math]\displaystyle{ \rho_f }[/math]. Poisson's equation is:

[math]\displaystyle{ \Delta\varphi=f }[/math]

and can also be written as:

[math]\displaystyle{ \nabla^2 \varphi = f. }[/math]

Electrostatics

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

[math]\displaystyle{ {\nabla}^2 \varphi = -\frac{\rho_f}{\varepsilon}. }[/math]

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.

The potential at a distance r from a point charge Q is:

[math]\displaystyle{ \varphi(r) = \dfrac {Q}{4 \pi \varepsilon r}. }[/math]

Connectedness

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.

See also

Electric Potential

Further reading

Books, Articles or other print media on this topic

External links

[1] [2]

References

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