Poisson equation Definition and 44 Threads

In the paper "Controlling the Electronic Structure of Bilayer Graphene", they did 5x10^(18) cm^-3 of nitrogen doping on a 6H-SiC substrate. They assumed a very thick graphite layer and a junction, and calculated the potential difference between the first and second layers of graphene near the...

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...

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...

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

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...

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

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, 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...

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: [SIZE="5"](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:##\nabla^2 u=-ku## is the same form of ##\nabla^2 u=f##. So is helmholtz equation a form of 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...

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 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, 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...

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...

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...

Am just playing around, and following examples of Fourier transform solutions of the heat equation, tried the same thing for the electrostatics Poisson equation \nabla^2 \phi &= -\rho/\epsilon_0 \\ With Fourier transform pairs \begin{align*} \hat{f}(\mathbf{k}) &= \frac{1}{(\sqrt{2\pi})^3}...

Hello everybody I've been searching this today but I am a bit lost now. I've encountered two forms of Gauss law in its differential form, Poisson equation : del2V(r) = -p(r)/e del2V(r) = -4*pi*p(r)/e where V:e.potential, p:charge density, e:permivity Now, what's the difference...

Hi Homework Statement Verify, that u(\vec{x}) := - \frac{1}{2 \pi} \int \limits_{\mathbb{R}^2} \log ||\vec{x} - \vec{y} || f(\vec{y}) d \vec{y} is the general solution of the 2 dimensional Poisson equation: \Delta u = - f where f \in C^2_c(\mathbb{R}^2) is...

In many book I read, problems for electrostatic potential always lead to solving Poisson equation. I saw a problem about a spherical shell carrying some amount of charges uniformly on the surface with density \rho, and then someone put a small patch on the sphere. The patch is then made a...

Hi everyone, I have been trying to solve both laplace and poisson equation using method of separation of variable but is giving me a hard time. Pls can anyone refer me to any textbook that solve this problem in great detail? Thanks

Suppose \phi is a scalar function: R^n\to R, and it satisfies the Poisson equation: \nabla^2 \phi=-\dfrac{\rho}{\varepsilon_0} Now I want to calculate the following integral: \int \phi \nabla^2 \phi \,dV So using Greens first identity I get: \int \phi \nabla^2 \phi \,dV = \oint_S \phi...

For a magnetostatics problem I seek the solution to the following equation \frac{1}{x}\frac{d}{dx} \left( x \frac{dy(x)}{dx} \right) = -C^2 y(x) (C a real constant) or equivalently x \frac{d^2 y(x)}{dx^2} + \frac{dy(x)}{dx} + C^2 x y(x)=0 It seems so simple, but finding a...

Homework Statement Consider a 1D sample, such that for x < xb the semiconductor has a dielectric constant \varepsilon_{1}, and for x > xb has a dielectric constant \varepsilon_{2}. At the interface between the two semiconductor matierials (x = xb) there are no interface charges. Starting...

1. Of what type is Poisson’s equation uxx + uyy = f(x,y) ? I used that if you have auxx+buxy+cuyy+dux+euy +fu+g=0 where a, b, c, d, e, f, g is constants, and if b^2-4ac<0 then you get an elliptic type because b=0, a=1, c=1 gives 0^2-4*1*1=-4<0 => elliptic Is this right? And why...