What is Laplace's equation: Definition and 116 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
∇
2
f
=
0
or
Δ
f
=
0
,
{\displaystyle \nabla ^{2}\!f=0\qquad {\mbox{or}}\qquad \Delta f=0,}
where
Δ
=
∇
⋅
∇
=
∇
2
{\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
f
(
x
,
y
,
z
)
{\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,
h
(
x
,
y
,
z
)
{\displaystyle h(x,y,z)}
, we have
Δ
f
=
h
.
{\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.
We have inhomogenous dirichlet boundary conditions (well understood)....the laplace equation is a steady state equation and we can clearly see that in 2D..it will be defined by 4 boundary conditions and NO initial condition...having said that; kindly have a look at the continuation below...
I...
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**...
I know the solution already, yet I don't understand why I'm able to solve laplace's equation when the volume has a point dipole within; I thought this required Poisson's eauation, which I have no experience in solving. Here is the relevant part of the solution
Hello everybody,
Currently I am doing my master's thesis and I've encountered a physics problem which is very difficult for me to solve. The problem I have is finding equations for the magnetic scalar potential inside and outside a ferromagnetic wire for specific boundary conditions...
Hi!
This thread might well be similar to:
https://www.physicsforums.com/threads/thread-about-jacksons-classical-electrodynamics-3rd-edition.910410/
I'm self-studying Vanderlinde and having a great time. However, I think that I am conflating and confusing many different things. Let me just ask...
I've tried to explicitly solve the Fourier's equation in cylindrical coordinates but I'm getting some messy integrals which cannot be solved analytically. Additionally my instructor said that there's a neat trick for this problem and it's possible to obtain the answer in a rather elementary...
I really have no idea as to how to attack the problem in the first place. I am here to ask for some generous help on how to start. The figure is shown below for reference.
Hi,
I have been learning about Laplace's equation recently, and have been wondering: how would we approach the problem if the region was a parallelogram (or some other shape that isn't a standard rectangle or circle)? Is this something that could feasibly be solved by hand, or would it require...
There seems to be a similarity between the solutions of laplace's equation and the principle of least action. e.g. the solution of a one dimensional laplace equation is a straight and the curve that minimizes the action is also a straight line. Was one derived from the other? Newbie here. Id...
Homework Statement
Prove that in a region free of electric charge, the value of the potential
at a point is equal to its average over any sphere centered at that point.
Homework Equations
V(r) = 1/(4piR^2) Integral(V * da)
The Attempt at a Solution
I defined a point outside the region where a...
I can find method of general solution of Laplace's equation in 3 D (in case cartesian, cylinder, and spherical coordinates)
From any book or any where ?
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...
Homework Statement
Hello,
I'm trying to solve laplaces equation to find a solution for the potential in a pipe with the given boundary conditions:
at x=b, V=V_0
at x= -b, V = -V_0
at y=a, V=0
at y=-a, V=0
(Assume this configuration is centered on the origin, pipe as dimensions -b<=x<=b...
Homework Statement
I'm trying to solve Laplace's equation numerically in 3d for a charged sphere in a big box. I'm using Comsol, which solves using the finite elements method. I used neumann BC on the surface of the sphere, and flux=0 BC on the box in which I have the sphere. The result does...
Homework Statement
-You are given a solution to Laplace's equation inside of a cylindrical region radius R.
-Show that by redefining the radial variable r as R2/r you get a solution for outside of R.
-Grounded conducting cylinder at r=R. Using a linear combination of the solutions in the...
A point charge q is situated at a distance d from a grounded conducting plane of infinite extent. Find the potential at different points in space.
I want to solve this problem without using the image charge idea.
I assumed azimuthal symmetry and took the zonal harmonics. And we know that as r...
This isn't homework but could be labeled "textbook style" so I'm posting it here.
Homework Statement
I'm trying to solve
\frac{\partial^2 u} {\partial x^2} +\frac{\partial^2 u} {\partial y^2}=0
on the domain x \in [-\infty,\infty], y\in[0,1] with the following mixed boundary conditions...
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...
I'm stuck on a seemingly simple 2D electrostatics problem. The problem is as follows:
A parabolic interface ($$x(y)=cy^2$$) separates two regions of different conductivities, with a uniform electric field at infinity aligned with the x-axis.
I write the Laplace operator in parabolic...
So here I have Laplace's equation with non-homogeneous, mixed boundary conditions in both x and y.
1. Homework Statement
Solve Laplace's equation \begin{equation}\label{eq:Laplace}\nabla^2\phi(x,y)=0\end{equation} for the following boundary conditions:
\phi(0, y)=2;
\phi(1, y)=0;
\phi(x...
Homework Statement
Consider solutions to the One Dimensional Laplace's Equation in Cartesian Coordinates
Let the range of x be from x1 to x2 (x1 > x2) and the boundary conditions are V[x1] = V1 and V[x2] = V2
Find the equation for V[x]
Homework Equations
V[x] = 1/2 (V(x+a)+V(x-a))
V[x] = mx...
Homework Statement
Essentially it gives the potential above the xy-plane as and I am tasked with verifying it satisfies laplace's equation, determining the electric field, and describing the charge distribution on the plane.
Homework Equations
then
The Attempt at a Solution
As far as I...
The angular equation:
##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=-l(l+1)\sin\theta\,\Theta##
Right now, ##l## can be any number.
The solutions are Legendre polynomials in the variable ##\cos\theta##:
##\Theta(\theta)=P_l(\cos\theta)##, where ##l## is a non-negative integer...
Homework Statement
I need to (computationally) solve the following linear elliptic problem for the function u(x,y):
\Delta u(x,y) = u_{x,x} + u_{y,y} = k u(x,y)
on the domain
\Omega = [0,1]\times[0,1] with u(x,y) = 1 at all points on the boundary.Homework Equations
[/B]
I know that I...
Homework Statement
Find the two-dimensional solution to Laplace's equation inside an isosceles right triangle. The boundary conditions are as is shown in the picture:
The length of the bottom and left side of the triangle are both L.
Homework Equations
Vxx+Vyy=0
V=X(x)Y(y)
From the image...
Homework Statement
A hollow cylinder with radius ##a## and height ##L## has its base and sides kept at a null potential and the lid on top kept at a potential ##u_0##. Find ##u(r,\phi,z)##.
Homework Equations
Laplace's equation in cylindrical coordinates...
Apparently,
f \nabla^2 f = \nabla \cdot f \nabla f - \nabla f \cdot \nabla f
where f is a scalar function.
Can someone please show me why this is step by step.
Feel free to use suffix notation.
Thanks in advance.
Hello,
Homework Statement
I am trying to solve Laplace's equation for the setup shown in the attachment, where f(x)=9sin(2πx)+3x and g(x)=10sin(πy)+3y. I have managed to solve it for the setup without the rectangle (PEC), and am now trying to solve ∇2\phi=0 for that inner rectangle in order...
In griffith's intro to electrodynamics (4rth edition), ch. 3, pg. 121.
here is the second uniqueness thrm for the solutions to laplace's equation:
the only part I'm confused about is, in the beginning where he says "in a volume V surrounded by conductors and containing a specified charge...
Hi,
Given a holomorphic function u(x,y) defined in the half plane ( x\in (-\infty,\infty), y\in (-\infty,0)), with boundary value u(x,0) = f(x) , the solution to this equation (known as the Poisson integral formula) is
u(x,y) = \int_{-\infty}^{\infty} \frac{y\ f(t) }{(t-x)^2 +y^2} \...
Homework Statement
Homework Equations
General solution:
Fourier series:
where r_{1}=a, r_{2}=b, f_{1}(\theta)=sin(\theta), and f_{2}(\theta)=2sin(\theta)cos(\theta).
The Attempt at a Solution
By evaluating the Fourier series shown above, I determined that...
"Two co-axial conducting cones (opening angles ##\theta_{1} = \frac{\pi}{10}## and ##\theta{2} = \frac{\pi}{6}##) of infinite extent are separated by an infinitesimal gap at ##r = 0##. If the inner cone is held at zero potential and the outer cone is held at potential ##V_{o}## find the...
Hi all,
Had a doubt regarding Laplace's equation.
In many textbooks, the general solution to the two dimensional Laplace's equation is mentioned as:
\Phi(\rho,\phi) = A_{0} + B_{0}ln(\rho) + \sum_{n=1}^{\infty}\rho^n(A_ncos(n\phi) + B_n sin(n\phi)) + \sum_{n=1}^{\infty}\rho^{-n}(C_n cos(n\phi)...
Homework Statement
A long rectangular metal plate has its two long sides and top at 0°. The base is at 100°. The plate's width is 10cm and its height is 30cm. Find the stead-state temperature distribution inside the plate.
Homework Equations
∇2T = 0
T(x,y) = X(x)Y(y)
X(x) = Acos(kx) +...
Homework Statement
Let ##U\subset\mathbb{R}^m## be a bounded set with smooth boundary ##\partial U##.
Consider a boundary value problem $$-\bigtriangleup u=f,\quad u|_{\partial U}=0.$$with ##f\in L^2(U)##.
Use the Riesz representation Theorem that the problem has a weak solution ##u\in...
Homework Statement
(From Plonsey, R. and R. C. Barr, "Bioelectricity: A Quantitative Approach")
Show
\nabla^2 r = 0
given
r = \sqrt{x^2 + y^2 + z^2}.
Homework Equations
\nabla = \frac{\partial}{\partial x}{\bf i} + \frac{\partial}{\partial y}{\bf j} +...
There seems to be a curious connection between Brownian Motion, stochastic diffusion process, and EM.
http://en.wikipedia.org/wiki/Stochastic_processes_and_boundary_value_problems
I was hoping to share and to have someone add some insight on on what it means that the Dirichlet boundary...
Homework Statement
This is a question related to finding the velocity field of an incompressible fluid in a square pipe with sides at y = ±(a/2) and x = ± (a/2).
It comes down to solving a homogenous equation which is also Laplace's equation
\frac {δ^2 w(x,y)^H}{δ x^2} + \frac {δ^2...
Homework Statement
Solve Laplace's equation on the rectangle 0< x< L, 0< y< H with the boundary conditions du/dx(0, y) = 0, du/dx(L, y)=y, du/dy(x, 0)=0, U(x, H)=x.
Homework Equations
The Attempt at a Solution
I would be able to solve it by separation of variables if the last...
Homework Statement
Rectangular pipe, infinite in the z direction. The sides in the y-z plane (at x=0 and x=a) are held at V=0, while the sides in the x-z plane (at y=0 and y=b) are held at V=V0
Explain why there cannot be a non-trivial solution to this configuration.
Homework...
Homework Statement
Use separation of variables to find the solution to Laplaces equation satisfying the boundary conditions
u(x,0)=0 (0<x<2)
u(x,1)=0 (0<x<2)
u(0,y)=0 (0<y<1)
u(2,y)= asin2πy(0<y<1)
The Attempt at a Solution
I am able to perform the separation of variables...
I'm trying to use conformal mapping to solve for a function u(x,y) satisfying Laplace's equation ∇2u = 0 on the outside of the unit circle (i.e. the complement of the unit disk), with boundary conditions:
u = 1 on the unit circle in the first quadrant,
u = 0 on the rest of the unit circle...
Homework Statement
The lecture notes say that ∇ = urr + (1/r)ur + (1/r2)uθθ. I'm not sure how this comes about. The notes never explain it.
Homework Equations
(?)
The Attempt at a Solution
No attempts on the actual homework problem until this ∇ thing is cleared up.
Homework Statement
Determine whether each of the following functions is a solution of Laplace’s
equation uxx + uyy = 0.
x^3 + 3xy^2
ux=3x^2
uxx=6x
uy=6xy^2
uyy=6x
6x+6x=12x and is therefore not a solution
Did I do that right? I'm just learning about this topic and it's...
Homework Statement
Suppose that T(x, y) satisfies Laplace’s equation in a bounded region
D and that
∂T/∂n+ λT = σ(x, y) on ∂D,
where ∂D is the boundary of D, ∂T/∂n is the outward normal deriva-
tive of T, σ is a given function, and λ is a constant. Prove that there
is at most one solution...