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.
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...
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...
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...
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 $$\Delta v=\Delta (u-\tilde{u})\Rightarrow -\Delta u + \Delta...
I tried to follow the method outlined in lectures, and ended up with an incorrect solution. My understanding of PDEs is a bit shaky so I thank anyone for constructive feedback or information. :bow:
The solution to the Poisson equation
\begin{equation}
-\Delta u(x)=\frac{q}{\pi...
We all know that Poissson's equation in electrostatic is:
$$\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...
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...
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)...
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}...
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...
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 ##...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...