Laplace equation Definition and 154 Threads

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)...

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...

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...

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...

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...

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...

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 learn that we can expand the electric potential in an infinite series of rho and cos(n*phi) when solving the Laplace equation in polar coordinates. The problem I want to consider is the expansion for the potential due to a 2D line dipole (two infinitely-long line charge separated by a small...

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates ##u=x+iy## and ##v=x-iy##. This can be extended to n dimensions as long as the complex coordinates chosen also solve the Laplace equation. For example in 3D...

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...

Suppose that $u$ is the solution of the Laplace equation $u_{xx}+u_{yy}=0$ in $\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$ $u(x,y)=x$ for all $(x,y)\in \mathbb{R}^2$ such that $x^2+y^2=1.$ Find the value of $u$ in $(0,0).$ Use the property of median value.

Let $\Omega$ an open domain in $\mathbb{R}^2.$ Suppose that $u$ is an application $C^2$ that satisfy the Laplace equation $\Delta u=0$ in $\Omega.$ Let $\Omega_{(a,b)}=\{(x+a,y+b): (x,y)\in \Omega\}$ and define $v(X,Y)=u(X-a,Y-b)$ for all $(X,Y)\in \Omega_{(a,b)}.$ Show that $v$ is an...

Hello everybody! I know how to solve Laplace equation on a square or a rectangle. Is there any easy way to find an analytical solution of Laplace equation on a trapezoid (see picture). Thank you.

Homework Statement Solve the Laplace equation inside a sphere, with the boundary condition: \begin{equation} u(3,\theta,\phi) = \sin(\theta) \cos(\theta)^2 \sin(\phi) \end{equation} Homework Equations \begin{equation} \sum^{\infty}_{l=0} \sum^{m}_{m=0} (A_lr^l + B_lr^{-l -1})P_l^m(\cos...

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...

1. The problem statement, all variables a nd given/known data A rectangular trough extends infinitely along the z direction, and has a cross section as shown in the figure. All the faces are grounded, except for the top one, which is held at a potential V(x) = V_0 sin(7pix/b). Find the...

< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >[/color] So this is the problem Here is the question: A 32lb weight strecthes a spring 2ft.The weight is released from rest at the equilibrium position. beginning at t=0, a force equal to f(t)= sint...

I have that ∇2∅ = 0 everywhere. ∅ is a scalar potential and must be finite everywhere. Why is it that ∅ must be a constant? I'm trying to understand magnetic field B in terms of the Debye potentials: B = Lψ+Lχ+∇∅. I get this from C.G.Gray, Am. J. Phys. 46 (1978) page 169. Here they found that...

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...

How do we know that separable solutions of Schrodinger equation (in 3d) form a complete basis? I understand that the SE is a linear PDE and therefore every linear combination of the separable solutions will also be a solution , but how do we know that the converse, i.e 'every solution can be...

I'm trying to solve Laplace equation using Fourier COSINE Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't think so). NOTE: U(..) is the Fourier Transform of u(..) This are the equations (Laplace...

I'm a bit confused wether or not there is a link between harmonic functions (solutions of the Laplace pde) and harmonic oscillating systems? What is the meaning of "harmonic" in these cases? Thanks!

Homework Statement use laplace transforms to solve the differential equation y"+2y'+17y = 1 Homework Equations Initial conditions are y(0) = 0 y'(0) = 0 The Attempt at a Solution so it converts to Y(s) (s^2+2s+17) = 1/s which then ends up as; Y(s) = 1/s*1/(s^2+2s+17) i know i need to invert...

Homework Statement This is example 3.9 in Griffiths Electrodynamics. "A specified charge density σ(θ) (inclination angle) is glued over the surface of a spherical shell of radius R. Find the resulting potential inside and outside the sphere." The problem suggests that although it is possible to...

So I'm currently taking electricity and magnetism and I'm expected to know how to perform a separation of variables on laplace equation in 2 dimensions.I have taken Zero differntial equations courses and I literally have no freaking idea what's going on. The book I use doesn't spend any time...

This question seems to come up often, but I cannot find a satisfying explanation. There is a point charge +Q some distance above an infinite conducting plane. Supposedly, the electric field below the plane must be zero. I have trouble understanding why this is true. The total charge on the...

The potential on the side and the bottom of the cylinder is zero, while the top has a potential V_0. We want to find the potential outside the cylinder. Can I use the same boundary conditions as for case of inside cylinder potential? What is different?

Homework Statement Write the Laplace equation \dfrac {\partial ^{2}F} {\partial x^{2}}+\dfrac {\partial ^{2}F} {\partial y^{2}}=0 in terms of polar coordinates. Homework Equations r=\sqrt {x^{2}+y^{2}} \theta =\tan ^{-1}(\frac{y}{x}) \dfrac {\partial r} {\partial x}=\cos \theta \dfrac...

Homework Statement I got to this while solving a physical problem, therefore it is hard for me to write the problem statement if there is none. I can write the whole physical problem here but it wouldn't make any sense. So, here is how it goes. I got to a Laplace equation ##\triangle \phi =0##...

Homework Statement i need to solve the laplace equation in square with length side 1 i tried to solve by superposition and i got infinite sum enen thouth i know that the answer should be finite Homework Equations 1.ψ(x=0,0≤y≤1)=0 2.ψ(y=0,0≤x≤1)=0 3.ψ(x=1,0≤y≤1)=10sin(∏*y)+3x...

Homework Statement Consider the Laplace’s equation, ∆u(r,θ) = 0, inside the quarter-circle of radius 2 (0 ≤ θ < π, 0 ≤ r ≤ 2), where the boundary θ is insulated, and u(r,\theta/2)=0 Show that the insulated boundary condition can mathematically be expressed as \frac{\partial u}{\partial...

Homework Statement Part(a): State condition for laplace's to work. Find potential in space between electrodes. Part (b): Find potential inside cylinder and outside. Part (c): How would the answer change is one pair is kept at potential 0 while other V0? Homework Equations The Attempt at a...

\nabla^2 u=\frac {\partial ^2 u}{\partial x^2}+\frac {\partial ^2 u}{\partial y^2}=\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} I want to verify ##u=u(r,\theta)##, not ##u(x,y)## Because for ##u(x,y)##, it...

The length of the side of the square is a. The boundary conditions are the following: (1) the left edge is kept at temperature T=C2 (2) the bottom edge is kept at temperature T=C1 (3) the top and right edges are perfectly insulated, that is \dfrac{\partial T}{\partial x}=0,\dfrac{\partial...

Homework Statement Heat equation in a annulus, steady state solution. u(a,θ,t) = Ta u(b,θ,t) = Tbcos(θ) Homework Equations Using separation of Variables \frac{}{}\frac{1}{r}\frac{d}{d r}(r\frac{d R}{d r}) + \frac{1}{r^2}\frac{d^2\Theta}{d \theta} = 0 The Attempt at a Solution...

Homework Statement Homework Equations Assume the solution has a form of: The Attempt at a Solution It looks like a sine Fourier series except for the 2c5 term outside of the series, so I'm not sure how to go about solving for the coefficients c5 and c10. Any idea?

Homework Statement The steady state temperature distribution T(x,y) in a flat metal sheet obeys the partial differential equation: \displaystyle \frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2} = 0 Seperate the variables in this equation just like in the...

Homework Statement An empty (charge-free) slab shaped region with walls parallel to the yz-plane extends from x=a to x=b; the (constant) potential on the two walls is given as Va and Vb , respectively. Starting with LaPlace's equation in one dimension, derive a formula for the potential at...

So we have the two ODE solutions are the cosine/sine and $r^n$ since it was a Cauchy Euler type. For the steady state, the solution is just a constant since it has to have period 2pi. But with $r^n$, how with lambda equal to zero does $\ln r$ come into play? If my question is hard to follow...

I have never solved an equation in polar form. I am not sure with how to start. Solve Laplace's equation on a circular disk of radius a subject to the piecewise boundary condition $$ u(a,\theta) = \begin{cases} 1, & \frac{\pi}{2} - \epsilon < \theta < \frac{\pi}{2} + \epsilon\\ 0, &...

Consider Laplace's equation $\nabla^2u = 0$ on the rectangle with the following boundary conditions: $$ u_y(x,0) = f(x)\quad u(L,y) = 0\quad u(x,H) = 0\quad u(0,y) = g(y). $$ How does one of the boundary conditions being defined by a derivative alter the solving of this problem? I have never...

Homework Statement Two semi-infinite conductor planes (like this ∠ ) have an angle β at a constant potential. Whats the potencial close to the origin? Homework Equations ∇²V = (1/r)(∂/∂r){ r ( ∂V/∂r ) } + (1/r²)(∂²V/∂\theta²) The Attempt at a Solution Trying for laplacian equation on...

[SIZE="4"]Homework Statement Show that If \phi(x,y,z) is a solution of Laplace's equation, show that \frac{1}{r}\phi (\frac{x}{r^2} ,\frac{y}{r^2} , \frac{z}{r^2} ) is also a solution Homework Equations The Attempt at a Solution let \psi= \frac{1}{r} \phi (\frac{x}{r^2}...

Good afternoon, I am a PhD student in motions of damaged ships. I am trying to find a solution of Laplace equation inside a box with a set of boundary conditions such that: ∇2\phi=0 \phix=1 when x=-A and x=A \phiy=0 when y=-B and y=B \phiz=0 when z=Ztop and z=Zbot I have tried...

Many texts in deriving the fundamental solution of the Laplace equation in three dimensions start by noting that the since the Laplacian has radial symmetry that Δu=δ(x)δ(y)δ(z) That all that needs to be considered is d^2u/dr^2 + 2/r du/dr = δ(r) For r > 0 the solution given is u= c1/r +...

Homework Statement ## \frac{\partial^2 V}{\partial x^2} +\frac{\partial^2 V}{\partial y^2} = 0 ## it's defined for ##V=0 ## for ## x=0, 0<y<b##, ## x=a, 0<y<b##, ## y=0, 0<x<a## ##V=V_0 x/a ## for ## y=b, 0<x<a## Homework Equations None. The Attempt at a Solution I've got ##V(x,b) =...

Hi, When can I assume that the solution the laplace equation (or poisson equation) is radial? That is, when can I look only at (in polar coordinates) \frac{\partial u^2} {\partial^2 r} + \frac{1}{r} \frac{\partial u} {\partial r} = f(r,\theta) instead of \frac{\partial u^2} {\partial^2 r}...

[FONT=tahoma]A metallic spherical shell occupying the region given in terms of spherical polar coordinates \( (r, \theta, \phi)\) by \(r \le a \) has its surface \(r = a\) maintained at temperature [FONT=CMR12] [FONT=tahoma]\(u(a,\theta)=1 + \cos(\theta)+2 \cos^2(\theta) \) [FONT=tahoma]...

Homework Statement Solve the Laplace equation in one dimension (x, i.e. (∂^2h)/(∂x^2)= 0) Boundary conditions are as follows: h= 1m @ x=0m h= 13m @ x=10m For 0≤x≤5 K1= 6ms^-1 For 5≤x≤10 K2 = 3ms^-1 What is the head at x = 3, x = 5, and x = 8? What is the Darcy velocity...

Homework Statement A square rectangular pipe (sides of length a) runs parallel to the z-axis (from -\infty\rightarrow\infty). The 4 sides are maintained with boundary conditions (i) V=0 at y=0 (bottom) (ii) V=0 at y=a (top) (iii) V=constant at x=a (right side) (iv) \frac{\partial...