Siméon Denis Poisson: Difference between revisions
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:<math>\Delta\varphi=f</math> | :<math>\Delta\varphi=f</math> | ||
where <math>\Delta</math> is the | 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 | ||
:<math>\nabla^2 \varphi = f.</math> | :<math>\nabla^2 \varphi = f.</math> | ||
==Electrostatics== | ==Electrostatics== | ||
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:<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 | 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. | ||
Using Green's Function, the potential at distance ''r'' from a central point charge ''Q'' (ie: the Fundamental Solution) is: | Using Green's Function, the potential at distance ''r'' from a central point charge ''Q'' (ie: the Fundamental Solution) is: | ||
Revision as of 17:54, 5 December 2015
Created by Benjamin Bierbaum
Siméon Poisson was a French mathematician best known for his work on definite integrals and electromagnetic theory.
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.
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.
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
Poisson's equation is
- [math]\displaystyle{ \Delta\varphi=f }[/math]
where [math]\displaystyle{ \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 ∇2 and so Poisson's equation is often written as
- [math]\displaystyle{ \nabla^2 \varphi = f. }[/math]
Electrostatics
Poisson's equation for electrostatics, which is:
- [math]\displaystyle{ {\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.
Using Green's Function, the potential at distance r from a central point charge Q (ie: the Fundamental Solution) is:
- [math]\displaystyle{ \varphi(r) = \dfrac {Q}{4 \pi \varepsilon r}. }[/math]
(For historic reasons and unlike gravity's model above, the [math]\displaystyle{ 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 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 charge density
- [math]\displaystyle{ \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]\displaystyle{ {\nabla}^2 \varphi = - { \rho_f \over \varepsilon } }[/math],
is given by
- [math]\displaystyle{ \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]\displaystyle{ {\nabla}^2 \varphi }[/math]. Note that, for r much greater than σ, the erf function approaches unity and the potential φ (r) approaches the point charge potential
- [math]\displaystyle{ \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 > 3σ the relative error is smaller than one part in a thousand.
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