What is Poisson equation: Definition and 62 Discussions

Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.

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  1. spin_100

    A Green's function for problems involving linear isotropic media

    I am considering a simple problem of a sphere of isotropic dielectric media (permittivity ## \epsilon ## and Radius ##R##) placed in a uniform electric field ## E_0 ## (z-direction). The problem is from Griffiths Chapter 4, example 7. Since, it is a linear dielectric material, ## D = \epsilon E...
  2. C

    I Where to find this uniqueness theorem of electrostatics?

    There is a nice uniqueness theorem of electrostatics, which I have found only after googling hours, and deep inside some academic site, in the lecture notes of Dr Vadim Kaplunovsky: Notice that the important thing here is that only the NET charges on the conductors are specified, not their...
  3. yucheng

    Confused about the nature of Laplace vs Poisson equation in BVP

    Hi! The problem clearly states that there is a surface charge density, which somehow gives rise to a potential. The author has solved the Laplace equation in cylindrical coordinates and applied the equation to the problem. So ##\nabla^2 V(r,\phi) = 0##, and ##V(a,\phi) = V_a(\phi)## (where...
  4. docnet

    Exploring the Structure of the Poisson Equation

    Theorem 1: There exists at most one solution ##u\in C^2(\bar{\Omega})## of the Dirichlet boundary value problem. Proof: (1) We assume there is a second solution ##\tilde{u}## of the Dirichlet boundary value problem. We compute $$\Delta v=\Delta (u-\tilde{u})\Rightarrow -\Delta u + \Delta...
  5. docnet

    Solving the Poisson equation with spherically symmetric functions

    I tried to follow the method outlined in lectures, and ended up with an incorrect solution. My understanding of PDEs is a bit shaky so I thank anyone for constructive feedback or information. :bow: The solution to the Poisson equation \begin{equation} -\Delta u(x)=\frac{q}{\pi...
  6. D

    A Solution of Poisson's Equation

    We all know that Poissson's equation in electrostatic is: $$\nabla^2\phi=-\frac{\rho}{\epsilon_0}$$ My question is: why the solution, let's say for 1D, is not just double integral as follows: $$\phi=\iint -\frac{\rho}{\epsilon_0} d^2x$$ which gives x square relation. But the actual solution...
  7. qnach

    Jackson Classical Electrodynamics: page 35 expansion of charge

    Could anyone explain how did Jackson obtain the Taylor distribution of charge distribution at the end of section 1.7 (version 3)?
  8. L

    Solving the 1D Poisson equation for a MOS device

    Hey everyone, I'm currently working on a 1D Poisson Solver for a MOS device (Al-Si-SiO2). Therefore, I programmed a Poisson Solver which is appling a boxintegration (Finite Volume Method) through the structure from φ(0) at the metal-oxide interface and φ(x_bulk = 20 nm) in in the silicon bulk...
  9. D

    Solving the 3D Poisson equation

    Hello ! I want to solve the 3D Poisson equation using spherical coordinates and spherical harmonics. First I must solve this : ##d^2\phi/dr^2 + 1/rd\phi/dr - l*(l+1)/r^2 = \rho (r)## with ##\phi (\infty ) = 0## (here ##\phi## is the gravitationnal potential and ##\rho## is the mass density)...
  10. K

    Calculate potential form poisson equation

    Hi. I've the following charge density: ## \rho = \rho_0 \frac {r}{R} ## I'm getting a trouble to calculate the potential inside a sphere of radius R located in the center of axis with the given charge density (using poisson equation): the Laplacian in spherical coordinates is: ##\frac {1}{r^2}...
  11. Alan Lins Alves

    Problem with the Finite Element Method applied to Electrostatics

    Hi! I have a code that solve the poisson equation for FEM in temperature problems. I tested the code for temperature problems and it works! Now i have to solve an Electrostatic problem. There is the mesh of my problem (img 01). At the left side of the mesh we have V=0 (potencial). There is a...
  12. C

    A Cylindrical Poisson equation for semiconductors

    In a cylindrical symmetry domain ## \Phi(r,z,\alpha)=\Phi(r,z) ##. Does anyone can point me what can be found in literature to solve, even with an approximate approach, this equation? \nabla^2 \Phi(r,z)=-\frac{q}{\epsilon} \exp(-\frac{\Phi(r,z)-V}{V_t}) Where ## q, \epsilon, V ## and ## V_t ##...
  13. A

    Classical Please recommend two textbookss about the Poisson equation and Green's function

    Please recommend two textbooks about Poisson equation, Green's function and Green's theorem for a theoretical physics student. One is easy to read so that I can have an overall understanding of the topics, another is mathematically rigorous and has a deep and modern exploration of these topics...
  14. V

    A Nanowire with charge neutrality in band gap

    I'm working on a finite element simulation of the electrostatic potential V(r) in and around semiconductor nanowires, based on solving Poisson's equation. While the details of the problem will follow shortly, the crux of where I run into trouble is that for nanowires it is important to include...
  15. A

    I Dimensional Analysis Poisson Equation

    Suppose I am given some charge density profile ρ(x). Poisson's equation in 1D reads d2φ/dx2 = ρ(x)/ε Is there a simple way to see what the order of magnitude of the electrostatic potential should be from a dimensional analysis?
  16. hikari1987

    A Exact solution to Poisson equation in 2D

    Hi all , Could you please help me solve Poisson equation in 2D for heat transfer with Dirichlet and Neumann conditions analytically? Thank you
  17. J

    I Solving the 3D Poisson Equation Using Finite Difference/Volume

    Hi, I'm attempting to solve the 3D poisson equation ∇ ⋅ [ ε(r) ∇u ] = -ρ(r) Using a finite difference scheme. The scheme is simple to implement in 3D when ε(r) is constant, and I have found an algorithm that solves for a non-constant ε(r) in 2D. But I am having trouble finding an algorithm...
  18. V

    I Poisson Equation Neumann boundaries singularity

    I am trying to solve the poisson equation with neumann BC's in a 2D cartesian geometry as part of a Navier-Stokes solver routine and was hoping for some help. I am using a fast Fourier transform in the x direction and a finite difference scheme in the y. This means the poisson equation becomes...
  19. S

    Poisson equation in R with a source at the origin

    Homework Statement Solve the poisson eq. on R with a source in x=0. The Attempt at a Solution I haven't done this kind of thing in years, so I'm a bit rusty, but I think that this is requested: \Delta \phi = - \rho \delta(x) (Edit: no wait, I need an integral here). It doesn't seem to be...
  20. S

    Boundary conditions for 3d current flow through water

    I've forgotten a lot of field theory so I've been rereading it in a couple of electric field theory textbooks. What seems like a simple problem falls between the cracks. I hope some readers can help - it will be appreciated. My application seems simple (solution will require numerical FEA but...
  21. Coffee_

    Poisson equation with a dirac delta source.

    Consider: ##\nabla^{2} V(\vec{r})= \delta(\vec{r})## By taking the Fourier transform, the differential equation dissapears. Then by transforming that expression back I find something like ##V(r) \sim \frac{1}{r}##. I seem to have lost the homogeneous solutions in this process. Where does this...
  22. T

    General Solution of a Poisson Equation of a magnetic array

    Hi, I'll give some background, say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0 and y = -d, respectively. The structure has scalar potentials inside it as so: As you can see the vector fields cancel out on one side, As it...
  23. F

    Poisson equation/zero padding and duplicating Green function

    Hello, I need to solve the Poisson equation in gravitational case (for galaxy dynamics) with Green's function by applying Fast Fourier Transform. I don't understand the method used for an isolated system from (Hockney & Eastwood 1981); it says : I have 2 questions: * Why we duplicate the...
  24. T

    General Solution of a Poisson Equation (maybe difficult)

    Hi, This is overwhelmingly more of a maths problem than a physics problem, because it's all theoretical. I'll give some background to modle it incase the math's isn't enough. Say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0...
  25. K

    Poisson equation for the field of an electron

    Homework Statement In classical electrodynamics, the scalar field \phi(r) produced by an electron located at the origin is given by the Poisson equation \nabla^2\phi(r) = -4\pi e\delta(r) Show that the radial dependence of the field is given by \phi(r) = \frac er Homework Equations I'm not...
  26. A

    Poisson equation with three boundary conditions

    I have the following 2D Poisson equation (which can also be transformed to Laplace) defined on a triangular region (refer to plot): \begin{equation} \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=C\end{equation} with the following three boundary conditions...
  27. R

    Poisson equation with finite difference method

    Hi guys , i am solving this equation by Finite difference method. (dt2/dx2 + dt2/dy2 )= -Q(x,y) i have developed a program on this to calculate the maximum temperature, when i change the mesh size the maximum temperature is also changing, Should the maximum temperature change with mesh...
  28. R

    MHB Boundary integral method to solve poisson equation

    Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential $V(r,z)$ with the help of changing the Poisson equation into cylindrical polar co ordinates? Where V is...
  29. E

    Solve the screened Poisson equation

    Homework Statement Solve the equation \nabla^2\phi-\frac{1}{\lambda^2_D}\phi=-\frac{q_T}{\epsilon_0}\delta(r) substituting the \delta representation \delta(r)=\frac{1}{4\pi}\frac{q_T}{r} and writing the laplacian in spherical coordinates. Use as your guess...
  30. Y

    Is wave and heat equation with zero boundary Poisson Equation?

    I have two questions: (1)As the tittle, if u(a,\theta,t)=0, is \frac{\partial{u}}{\partial {t}}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2} and \frac{\partial^2{u}}{\partial...
  31. Y

    Question on using solution from Helmholtz equation in Poisson equation

    Helmholtz equation stated that \nabla^2 u(r,\theta,\phi) =-ku(r,\theta,\phi) = f(r,\theta,\phi) This is being used for Poisson equation with zero boundary: \nabla^2 u(r,\theta,\phi) = f(r,\theta,\phi) and u(a,\theta,\phi)=0 I just don't see how this can work as k=m^2 is a number only...
  32. Y

    Is Helmholtz equation a Poisson Equation?

    Helmholtz equation:##\nabla^2 u=-ku## is the same form of ##\nabla^2 u=f##. So is helmholtz equation a form of Poisson Equation?
  33. maverick280857

    Basic question about the generalized Poisson Equation

    Hi, Suppose we look at two dimensional Poisson's equation in a medium with spatially varying (but real) dielectric constant: \nabla(\epsilon_r\nabla \varphi) = -\frac{\rho(x,y)}{\epsilon_0} Consider the problem of solving this using the Finite Difference method on a rectangular grid...
  34. C

    2-D Poisson Equation Boundary Value Prob

    Homework Statement Solve the equation: ∂2F/∂x2 + ∂2F/∂y2 = f(x,y) Boundary Conditions: F=Fo for x=0 F=0 for x=a ∂F/∂y=0 for y=0 and y=b Homework Equations How can I find Eigengunctions of F(x,y) for expansion along Y in terms of X? The Attempt at a Solution I can't imagine...
  35. F

    Complex form of poisson equation

    Hi, I am trying to solve the complex form of poisson equation ∇[ε(x)∇V(x)] = -1/εo ρ(x) with complex permittivity. If I introduce complex permittivity ε = εr - j σ/(εow), then I must introduce a complex potential ,V = Vr + jVi. That means the charge density, ρ, must also take a complex...
  36. B

    Green's function for Poisson Equation

    Hi, I am working on finding a solution to Poisson equation through Green's function in both 2D and 3D. For the equation: \nabla^2 D = f, in 3D the solution is: D(\mathbf x) = \frac{1}{4\pi} \int_V \frac{f(\mathbf x')}{|\mathbf x - \mathbf x'|} d\mathbf{x}', and in 2D the solution is: D(\mathbf...
  37. A

    Uniform Field & Poisson equation Mismatch?

    Hi, I'm getting some confusing results and can't figure out what is wrong Suppose we have a uniform field E=[0,0,E_z] in a dielectric media. By E=-\nabla\psi we can deduce that \psi(x,y,z)=-z E_z But, taking the Laplacian \nabla^2\psi=\frac{\partial^2 (-zE_z)}{\partial z^2}=0...
  38. S

    Solving Poisson Equation by using FDM

    I need help from anyone urgently, I need C code for Solving Poisson Equation has known source with Neumann condition by using FDM (finite difference method) in 2D problem.
  39. L

    How Einstein field equation becomes the Poisson equation?

    I want to show that ∇2ϕ=ρ/2, which governs gravity in Newtonian physics? I found solution of this question in [General Relativity for Mathematicians, R.K.Sachs and H.Wu, 1997, page 112&271]. Solution refer to optional exercise as follows: Let R^ be the (0, 4) –tensor field physically...
  40. P

    MATLAB Poisson Equation with Mixed Boundary Condition - MatlaB

    Hi guys, I'm solving a Poisson Equation with Mixed Boundary condition. But I have trouBle with that mixed BC in MATLAB. Anyone can help to fix? Thanks a lot! dT^2/dx^2+dT/dy^2=-Q(x,y)/k Rectangular domain (HxL), BC: Top: T(x,H)=Th, Left: dT/dx=0, Bottom: dT/dy=q, Right: dT/dx+B(T-Tinf)=0...
  41. M

    Poisson Equation for a Scalar Field

    We all know that for the gravitational field we can write the Poisson Equation: \nabla^2\phi=-4\pi G\rho But I was wondering if, mathematically, we can write the same equation for a scalar field which scale as r^{-2}. Here is the thing. When you deal with gravity, the Poisson equation is...
  42. B

    Standard Benchmark Problem for Computational Solution of Poisson Equation

    Hi, I am working on FEM methods as a part of my senior year project and I have written a poisson solver for the same purpose. The solver works pretty well on the simple problems that I have designed as of now and seems to give correct answer (i.e. the data matches the theoretical prediction) 1...
  43. Z

    FEM software? Poisson equation: Boundary conditions for a charged boundary

    Hi all, I want to calculate the electrostatic potential for an two-dimensional area with given Dirichlet boundary conditions (say, a square) with a charged ring in it (like a wedding ring, but inifinitely thin) with a given line charge density. I figured out that the problem should be...
  44. H

    Gaussian Elimination Solution to the 2D Poisson Equation

    I am trying to use Gaussian elimination to solve the 2D poisson equation. I've done this for the 1D problem without problems, but for some reason my solution for the 2D problem is incorrect; it looks something like the correct solution but it's as if the resulting field were cut in half, so...
  45. R

    Poisson equation: what shape gives largest area average

    When considering the solution u(x,y) of the poisson equation u_xx + u_yy = -1 for (x,y) in G on a 2-dimensional domain G with Dirichlet boundary conditions u = 0 for (x,y) on boundary of G I am wondering the following: for what shape of the domain G do I obtain the largest area-average...
  46. O

    Solving fluid's Poisson equation for periodic problem or more easy way?

    The problem is about mathematics but it originates from the self-gravitational instability of incompressible fluid, so let me explain the situation first. I have an incompressible uniform fluid disk that is infinite in the x-y direction. The disk has a finite thickness 2a along the...
  47. T

    Help with Poisson equation in irregular mesh

    Hi all, I've written a program that solves the Poisson equation in an irregular mesh, using a finite volume discretization. the method works well, and the solution is very good at a distance of 2-3 mesh sizes off of the source. The problem is that I need to know what is the contribution of...
  48. A

    Stiff- One-Dimensional Poisson Equation in Plasma

    Hey Guys, So I am trying to model the development of a collisional plasma in time. Now the problem I face is at the sheath boundary the changes in the charge densities is very large. I use the charge densities to evaluate the electric potential at different points in the plasma. I have...
  49. Y

    Poisson equation general solution

    Homework Statement Given that \nabla2 1/r = -4\pi\delta3(r) show that the solution to the Poisson equation \nabla2\Phi = -(\rho(r)/\epsilon) can be written: \Phi(r) = (1/4\pi\epsilon) \int d3r' (\rho(r') / |r - r'|) Homework Equations The Attempt at a Solution I know...
  50. Z

    How can I solve a Dirichlet problem defined for a parabolic region?

    Homework Statement Hi, I am looking for a hint, how to solve the following Dirichlet problem. All the standard textbooks have only examples for Dirichlet problems in rectangular or polar coordinate systems, but this problem is defined for a parabolic region. Homework Equations uxx+uyy=2...