All Questions

43 questions
21 views

Laplace equation outside sphere

When solving the Laplace equation on sphere coordinates you get: $$u(r,\theta) = \sum_{n=0}^{\infty}\left( A_n\,r^n + \frac{B_n}{r^{n+1}} \right) P_n(\cos\theta)$$ And it is clear that if you have ...
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Electric potential and field due to a continuous charge distribution

(1) The electric potential due to a continuous charge distribution is: $$\psi=\int_V \dfrac{\rho}{r}\ dV$$ To calculate this integral $\rho$ must be continuous over $V$. But $\rho$ is discontinuous ...
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Infinite annular potential well. Trouble with solving Bessel equation to get eigenstates and energy

I have infinite annular potential well (scheme in the picture). Schrodinger equtation in the anullus (for $R_1 <r<R_2$ is $V=0$) with polar coordinates is \begin{equation} - \frac{ \hbar }{...
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Behavior of potential by infinite charge distribution

The picture is a question from the book Intro to Electrodynamics by Griffiths. In question as you can see we want to find potential due to an infinite strip maintained at constant potential in the ...
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Linear, homogenous and isotropic dielectric in electrostatic field. Why do I consider two potentials (inside & outside sphere)?

Presentation of the problem : We have a uniform homogenous isotropic dielectric sphere in an electrostatic field. To solve this problem, we remark that we have an azimuthal symmetry. So the ...
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$\sin$ and $\cos$ components in symmetric infinite potential well problem

Consider an infinite potential well in one dimension with boundaries at $\pm a/2$. Can $\psi(x) = A \sin(kx) + B \cos(kx)$ for this system? The way it was answered was "mathematically acceptable but ...
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Charge on a conductor's surface

I take a charged conductor completely insulated. The charge is distributed over the surface, maintaining the surface at a given potential. The charge distribution that gives this potential is unique?
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A point charge near an infinite conducting plane

I want to calculate (with Poisson's equation) the electric field in the region containing a point charge near an infinite conducting plane with 0 potential. My textbook uses V(x,y,z)= 0 for every x,y,...
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How do I show that the Laplace equation has a unique solution under the Dirichlet closed-surface boundary condition?

In other words, when the the potential is specified at a finite boundary, how can I show the solution to $\nabla ^{2} V = 0$ exists and is unique? It is fine to show it for two dimensional Cartesian ...
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Trouble applying boundary conditions to Laplace equation

I'm having trouble finding which conditions to apply to Laplace equation in order to find the electrostatic potential of a specific configuration: There are 4 electrodes, given by the equations (each ...
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Solving the TISE for Infinite square well mathematical question

Consider the infinite square well situation where the potential is infinite at positions $|x| > a$ and $0$ otherwise. When solving the Time independent Schrodinger Equation (TISE) we can come to ...
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Energy spectrum for a step potential

Most of the books tend to give this explanation that for a bound physical system, the energy and momentum eigen values have discrete spectrum and otherwise, they have a continuous spectrum, which I ...
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An electrostatic problem for two disks in $\mathbb{R}^2$ - how can the solution be represented?

The electrostatic Laplace problem for the exterior of a disk can be solved in a straightforward manner using separation of variables. Suppose we have a unit disk $\Omega$ with a charge density of $f$ ...
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Dielectric cylinder in uniform electric field: nonlong cylinder

Problem of dielectric cylinder in uniform electric field is well known. For example, Jackson textbook or Griffiths textbook or online solution here. Solution always given for case of long cylinder. ...
601 views

Intuition in infinite grounded conducting plane with a point charge above it

In this problem, we use the Method of Images and get the resulting properties of the charge distribution on the plane. At the end of this process, Griffiths, in Introduction to Electrodynamics, ...
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Dipole field issue in particle-mesh Ewald method with periodic boundary conditions

I am working on a thesis that makes a great use of molecular-dynamics simulations, and I am trying to understand how the particle-mesh Ewald method works. The problem is, I have difficulties ...
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Choosing $A_l=0$ to guarantee bounded potential in infinity

I'm taking a course in Electrodynamics and quite often, when using the spherical approach $$\Phi=\sum\limits_{l~=~0}^{\infty}\left(A_lr^l+B_lr^{-(l+1)}\right)P_l(\cos\gamma),$$ there's the argument ...
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