What is Laplace equation: Definition and 155 Discussions

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as





{\displaystyle \nabla ^{2}\!f=0\qquad {\mbox{or}}\qquad \Delta f=0,}




{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}
is the Laplace operator,

{\displaystyle \nabla \cdot }
is the divergence operator (also symbolized "div"),

{\displaystyle \nabla }
is the gradient operator (also symbolized "grad"), and


{\displaystyle f(x,y,z)}
is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
If the right-hand side is specified as a given function,


{\displaystyle h(x,y,z)}
, we have


{\displaystyle \Delta f=h.}
This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.
The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.

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  1. Harikesh_33

    I Question regarding Laplace's Equation for regions with charges

    Why doesn't the **Laplace's equation**(#\nabla^2V=0#) hold in the region within the sphere when there is a charge inside it ? Is it because #ρ \ne 0# within the sphere and it becomes a **poisson equation**($\nabla^2V=\dfrac{-ρ}{ε_0}$) and changes the characteristics of **Harmonic Solution**...
  2. P

    I Laplace equation with irregular boundaries

    Is there a way to solve Laplace’s Equation on irregular domains if the domain’s shape is given by a function for example a 2D parabolic plate. I keep seeing numerical methods but I want to know is there an ANALYTICAL method to solve it on an irregular domain. If there isn't are there approximate...
  3. Ahmed1029

    I Laplace's equation in presence of a dipole (perfect or physical)

    Does Laplace's equation hold true for electrostatic potential at the location of a dipole? Or should poisson's equation be used?
  4. 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...
  5. Rlwe

    Capacitance of infinitely long coaxial cylinders of elliptical section

    I've been able to prove the following inequality $$\frac{2\pi\epsilon_0}{\log\left(\frac{b_1b_2}{a_1^2}\right)}\leq C \leq \frac{2\pi\epsilon_0}{\log\left(\frac{a_1a_2}{b_1^2}\right)}$$ but have no clue how to obtain exact value. Can someone check whether this inequality is correct and show how...
  6. L

    Fourier transform to solve this Laplace equation

    I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result: $$ \tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky} $$ However, I'm having some problems with the inverse transform: $$ \frac{1}{2\pi}\int_{-\infty}^\infty...
  7. J

    Using the Frobenius method on a 2D Laplace

  8. jawad hussain

    Laplace Equation Numerical Solution

    I wonder how to incorporate point charge?
  9. G

    A Laplace eq. in cylindrical coordinates and boundary conditions

  10. docnet

    Radial solutions to the Laplace equation

    Part 1 $$\Delta u(x)=\Delta v(|x|)$$ Substitute $$|x|=r=\sqrt{\sum_{i=1}^n{x^2_i}}$$ $$u'(x)= v'(r)\frac{\sum_{i=1}^nx_i}{\sqrt{\sum_{i=1}^n{x^2_i}}}$$ $$u''(x)=v''(r)\frac{\sum_{i=1}^nx_i}{\sqrt{\sum_{i=1}^n{x^2_i}}}+v'(r)f(x)=v''(r)+v'(r)f(x)$$...
  11. Z

    Tackling Boundary Conditions in Python (Griffins Example)

    How to run a numerical simulation of Laplace equation if one of the boundary condition is like this: $$V(x,y) = 0 \text{ when } x \to \infty$$ I am trying to use Python to plot the solution of this Example 3.5. in Griffins EM
  12. F

    Show that the real part of a certain complex function is harmonic

    Hello, I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) $$ is harmonic on the whole complex plane. This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend...
  13. K

    Charged Bubble Oscillation

    From Gauss's Law give ##E=\dfrac{\sigma}{2\epsilon_0}## ##\therefore P_e=\dfrac{\sigma^2}{2\epsilon_0}## Consider at equilibrium (before bubble being charged) ##P_i=P_0+\dfrac{4S}{R}## Using Newton's 2nd Law ##\Sigma F=m\ddot{R}## Let ##R+\delta R## be the new radius Give (after binomial...
  14. person123

    Lines of Stress in a Material

    I was initially curious by the fact that streamlines around a circle appear the same as the lines of stress around a hole: I understand that streamlines are the contour lines of the stream function ##\psi## which satisfies the Laplace equation. I was wondering there is a related function for...
  15. L

    Engineering Help with Homework: Solving a Math Formula

    i want to ask about my homework im not understand what to do with this formula :
  16. C

    Electrical potential of a thin wire in an E field

    Assume that an infinite metallic plate A lies in the xy-plane, and another infinite metallic plate B is parallel to A and at height z = h. The potential of plate A is 0, and the potential of plate B is constant and equal to V. So, there is a uniform electrostatic field E between plates A and B...
  17. migueldbg

    Potential from point charge at distance ##l## from conducting sphere

    After looking around a bit, I found that, considering the polar axis to be along the direction of the point charge as suggested by the exercise, the following Legendre polynomial expansion is true: $$\begin{equation}\frac{1}{|\mathbf{r} - \mathbf{r'}|} = \sum_{n=0}^\infty...
  18. F

    I Separation of Variables (PDE) for the Laplace Equation

  19. dRic2

    Simple electric potential and Laplace equation

    Imagine to be in 2 dimensions and you have to find the potential generated by 4 point-charges of equal charge located at the four corners of a square. To do that I think we simply add all the contributions of each single charge: $$V_i(x, y) = - \frac k {| \mathbf r - \mathbf r_i|}$$ $$ V(x, y)...
  20. majormuss

    Visualizing & Solving a 2D Laplace Eq problem (Polar Coordinates)

  21. Arman777

    The mean value of the cube, Force Field Laplace equation

    Homework Statement I have a value of $$ U=U_0+x (∂U/∂x)+y(∂U/∂y)+z (∂U/∂z)+1/2x^2(∂^2U/∂x^2)+1/2y^(2∂^2U/∂y^2)+...$$ We need to find the mean value of the U. So the answer is $$\overline{\rm U}\approx U_0+a^2/24(∇^2U)$$ Homework Equations $$\overline{\rm U}=1/a^3 \int \int\int Udxdydz$$...
  22. CptXray

    Laplace equation in a cube

    Homework Statement There's a metal cunducting cube with edge length ##a##. Three of its walls: ##x=y=z=0## are grounded and the other three walls: ##x=y=z=a## are held at a constant potential ##\phi_{0}## . Find potential inside the cube. Homework Equations The potential must satisfy Laplace...
  23. M

    Mathematica Solving the Laplace Equation in weird domains

    Hi PF! I looked through the documentation on their website, but under the tab "Solve partial differential equations over arbitrarily shaped regions" I am redirected to a page that does not specify how to create a region. Any help is greatly appreciated. Also, if it helps, the domain is a...
  24. md nabil

    A Solving the Laplace equation over a trapezoidal domain

    can anyone help me on how I can map an isosceles trapezoid onto a rectangular/square domain.Actually I need to solve Laplace equation(delta u = 0) over this isosceles trapezoidal domain. Schwarz Christoffel mapping may help me. But can anyone give me any hint on this mapping procedure?
  25. evinda

    MHB Solve Laplace equation on unit disk

    Hello! (Wave) I want to solve the Laplace equation on the unit disk, with boundary data $u(\theta)=\cos{\theta}$ on the unit circle $\{ r=1, 0 \leq \theta<2 \pi\}$. I also want to prove that little oscillations of the above boundary data give little oscillations of the corresponding solution of...
  26. G

    A solution to the Laplace equation

    Hi, I am looking for the solution to the quadrant problem of the Laplace equation in 2 dimensions with Dirichlet boundary conditions \begin{equation} \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \end{equation} in the first quadrant ## x, y \geq 0 ## with boundary...
  27. evinda

    MHB Boundary value problem for Laplace equation

    Hello! (Wave) Let $a,b>0$ and $D$ the rectangle $(0,a) \times (0,b)$. We consider the boundary value problem in $D$ for the Laplace equation, with Dirichlet boundary conditions, $\left\{\begin{matrix} u_{xx}+u_{yy}=0 & \text{ in } D,\\ u=h & \text{ in } \partial{D}, \end{matrix}\right.$...
  28. Selveste

    Solve the Laplace equation in 2D

    Homework Statement Solve the Laplace equation in 2D by the method of separation of variables. The problem is to determine the potential in a long, square, hollow tube, where four walls have different potential. The boundary conditions are as follows: V(x=0, y) = 0 V(x=L, y) = 0 V(x, y=0) = 0...
  29. B

    I Can the Schrodinger equation satisfy Laplace's equation?

    The time-dependent Schrodinger equation is given by: ##-\frac{\hslash^{2}}{2m}\triangledown^{2}\psi+V\psi=i\hslash\frac{\partial }{\partial t}\psi## Obviously, there is a laplacian in the kinetic energy operator. So, I was wondering if the equation was rearranged as...
  30. H

    A Applying boundary conditions on an almost spherical body

    I am solving the Laplace equation in 3D: \nabla^{2}V=0 I am considering azumuthal symmetry, so using the usual co-ordinates V=V(r,\theta). Now suppose I have two boundary conditions for [V, which are: V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad...
  31. F

    Electric potential outside an insulator in a uniform field

    Homework Statement An Ohmic material with some conductivity has a uniform current density J initially. Let's say the current is flowing in the direction of the z-axis. A small insulating sphere with radius R is brought inside the material. Find the potential outside the sphere. Homework...
  32. lightest

    Point charge with grounded conducting planes angled 120

    Homework Statement The problem states: "A point charge q is located at a fixed point P on the internal angle bisector of a 120 degree dihedral angle between two grounded conducting planes. Find the electric potential along the bisector." Homework Equations ΔV = 0 with Dirichlet boundary...
  33. stockzahn

    A Laplace Equation for an annular geometry with different BC

    Dear all, I would need mathematical help to solve for the temperature field in an annular geometry (you find a picture attached below the text): A copper pipe containing a boiling two-phase flow (in the stratified regime) is immersed in a liquid bath, which temperature ##T_{IY}## is assumed to...
  34. V

    Find potential between 2 conc. cyl. with grounded strip

    Homework Statement Two concentric cylinders with radii a & b (b>a) with an infinitely long grounded strip along the z-direction are given potentials \phi_1 and \phi_2. Find \Phi(r,\phi) for a<r<b Boundary conditions: \Phi(r,2n\pi)=0 \Phi(a,\phi)=\phi_1 \Phi(b,\phi)=\phi_2 Homework...
  35. M

    A Young Laplace Equation

    Hi PF! I am considering a partial cylinder filled with fluid. By partial I mean consider something like a half-pipe. If a small disturbance is present, the fluid radius on the open side of the cylinder is ##r=R(1+\epsilon f(\theta,z,t))##. The Young-Laplace equation for capillary pressure is...
  36. F

    Stationary temperature field inside a house

    Homework Statement House: a room (see figure) has perfectly isolated walls, except the two windows where a convective heat exchange takes place (with the same transfer coefficient). Outside temperature in front of a sun-faced wall-sized panoramic window is T1, while at the back it is...
  37. C

    A Laplace equation- variable domain

    Hi, I need to solve Laplace equation ## \nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ## r_1<r<r_2 ##, ## 0<z<L ##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ &\Phi(L,r)=V_P \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial...
  38. H

    A Problems with identities involving Legendre polynomials

    I am studying the linear oscillation of the spherical droplet of water with azimuthal symmetry. I have written the surface of the droplet as F=r-R-f(t,\theta)\equiv 0. I have boiled the problem down to a Laplace equation for the perturbed pressure, p_{1}(t,r,\theta). I have also reasoned that...
  39. R

    Boundary conditions in dielectric problems

    Q) A conducting sphere of radius R floats half submerged in a liquid dielectric medium of permittivity e1. The region above the liquid is a gas of permittivity e2. The total free charge on the sphere is Q. Find a radial inverse-square electric field satisfying all boundary conditions and...
  40. C

    I Laplace equation boundary conditions

    Hi, I need to solve Laplace equation ##\nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ##r_1<r<r_2##, ##0<z<L##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ &\Phi(L,r)=V_P \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial...
  41. maistral

    A 2D Cartesian Laplace equation with a single point diffusion

    Hi. I have this problem in trying to solve this PDE analytically. The PDE is represented by this diagram: Basically this is solving the Laplace equation with those insulated boundaries except it has that point diffusing its value across the plane. I know how to solve the Laplace equation...
  42. S

    Laplace equation

    Consider an infinitely long hollow dielectric cylinder of radius a with the electricpotential V = V0 cos φ on the surface of the cylinder where φ is an angle measured around the axis of the cylinder. Solve Laplace’s equation to find the electric potential everywhere in space. Do you just plug V...
  43. Dopplershift

    Partial Differentiation Laplace Equation Question

    Homework Statement Consider the Laplace Equation of a semi-infinite strip such that 0<x< π and y>0, with the following boundary conditions: \begin{equation} \frac{\partial u}{\partial x} (0, y) = \frac{\partial u}{\partial x} (0,\pi) = 0 \end{equation} \begin{equation} u(x,0) = cos(x)...
  44. Alvis

    I Complex Analysis Harmonic functions

    Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic. I tried using the Laplace Equation of Uxx+Uyy=0 I have: du/dx=Ux d^2u/dx^2=Uxx du/dy=Uy d^2u/dy^2=Uyy dv/dx=cVx d^2v/dx^2=cVxx dv/dy=cVy d^2v/dy^2=cVyy I'm not really sure how to...
  45. D

    I Kernel of Laplace equation

    Hi! Everyone. I encounter some trouble in deriving the kernel of Laplace equation with prescribed boundary conditions. Given the following preposition: $$T(x, y) = \int_{-\infty}^{\infty}dx'\frac{y/\pi}{(x-x')^{2}+y^2}F(x')...[1]$$ satisfies the Laplace equation for ##x\in(-\infty, \infty)##...
  46. T

    Problem with solving laplace equation with a charged ring

    Homework Statement A ring charge of total charge Q and radius a is concentric with a grounded conducting sphere of radius b, b < a. Determine the potential everywhere. The ring is located in the equatorial plane, so both the sphere and the ring have their center at the same spot. Homework...
  47. Conservation

    Total thermal energy from heat equation

    Homework Statement Homework Equations Heat equation The Attempt at a Solution I can derive E(t) to get integral of du/dt over 0 to L, which is the same as integrating the right hand side of the original equation (d2u/dx2+sin(5t); while this allows me to take care of the d2u/dx2, I don't know...
  48. T

    Magnetic Mirror for Neutral Atoms

    Homework Statement Consider an infinite sheet of magnetized tape in the x-z plane with a nonuniform periodic magnetization M = cos(2πx/λ), where λ/2 is the distance between the north and south poles of the magnetization along the x-axis. The region outside the tape is a vacuum with no currents...
  49. Dor

    Solve Laplace equation on rectangle domain

    Homework Statement I'm having issues with a Laplace problem. actually, I have two different boundary problems which I don't know how to solve analytically. I couldn't find anything on this situations and if anybody could point me in the right direction it would be fantastic. It's just Laplace's...
  50. DrPapper

    Potential of a Rectangular Pipe by Laplace's Equation

    Homework Statement Homework Equations Is my part a correct and am I on the right track for part b? If not please give me some suggestions to get me closer to the right track. Also, how would I even begin c.? We have literally done no examples like this in class. The Attempt at a Solution...