4
votes
2answers
55 views

Conductors and Uniqueness Theorem

I'm working with Griffiths Electrodynamics, and he introduces a uniqueness theorem: First Uniqueness Theorem: The potential $V$ in a volume $\Omega$ is uniquely determined if (a) the charge ...
1
vote
0answers
34 views

Contradictory boundary conditions in electrostatics problem?

Consider the following problem: A conducting cube of side $a$ is grounded. Inside there's a horizontal (i.e., perependicular to the $z$ axis) sheet with uniform surface charge density $\sigma$. The ...
1
vote
1answer
53 views

Does charge distribute itself uniformly on a conductor?

An excerpt from a beginning E&M book [...] In other words, the surface of a conductor is an equipotential surface under static conditions. [...] Summarizing the boundary conditions at the ...
2
votes
1answer
99 views

Laplace's Equation - under what circumstances does it hold?

I'm currently taking an EM course whereby we deal with systems that satisfy Laplace's equation $\nabla^2 \phi = 0$. Examples include permeable sphere in a magnetic field and metal sphere in electric ...
2
votes
1answer
116 views

Boundary conditions in Electrostatics

If I have a grounded conducting material, then I know that $\phi=0$ inside this material, no matter what the electric configuration in the surrounding will be. Now I have a conducting material that ...
6
votes
1answer
102 views

Green function two solutions questions

I am having some trouble with Green functions in electrostatics What is the meaning of this trick: Given $$\vec{\nabla}^2 V(\vec{r}) = \frac{-1}{\varepsilon_0}\rho(\vec{r}) = ...
3
votes
1answer
119 views

Solving non-linear ODE for divalent solution at a 1-D surface boudary

I am trying to solve the following equation for a positively charged plane with charge density $\sigma$ at $z = 0$. $$ \phi''(z)=-\frac{e}{\epsilon \epsilon_0} \big(z_+n_{+} e^{-\beta z_+ ...
2
votes
1answer
148 views

An Electric Potential Glued to a Cube-Shaped Insulator to Replicate a Point Charge: Charge Distribution

I have been going back over this problem with a friend for the better part of a day: A potential is glued to a cube-shaped insulator so that outside of the insulator the field is the same as a point ...
2
votes
0answers
77 views

Equivocal boundary conditions for Laplace equation of 2D “V”-shape conductor

A two dimensional infinite "V"-shape wedge conductor is earthed, wherein $\beta$ is the intersection angle. We can solve Laplace equation so as to get the electric potential inside the "V" zone, as is ...
3
votes
1answer
106 views

Is the system of equations of electrostatics underdetermined or overdetermined? [duplicate]

The following equations are equations of electrostatics: $$\nabla \times \vec E=0$$ $$\nabla\cdot\vec E=\dfrac{\rho}{\epsilon_0}.$$ These are 4 independent equations, while $\vec E$ has only 3 ...
0
votes
1answer
729 views

Boundary condition for a floating electrostatic potential

I have a (probably) simple question regarding boundary conditions. In electrostatic simulations, the relevant Maxwell equation is $\nabla \cdot \mathbf{D}=\rho$ where $\mathbf{E}=-\nabla V$, and ...
-1
votes
2answers
600 views

Meaning of boundary condition for steady current density?

Although I understand the derivation of boundary condition in case of steady electric current but I did not understand, that the electric field which is in direction of $J$ current density that is ...
2
votes
1answer
105 views

Behavior of the electric field on boundary surfaces

Consider this picture. Integrating over this infinitesimal box gives the following equivalencies: $$\int_{\Delta V} d^3r~{\rm div} \vec{E}(\vec{r}) = \int_{S(\Delta V)} d\vec{f} \cdot ...
2
votes
1answer
174 views

Boundary conditions for Laplace's equation

Given a grounded conducting sphere, $V=0$ and $radius = R$, centered at the origin with a pure electric dipole (dipole moment $\vec p$) situated at the origin and pointing along the positive $z$ axis, ...
2
votes
2answers
8k views

Dirichlet and Neumann Boundary condition: physical example

Can anybody tell me some practical/physical example where we use Dirichlet and Neumann Boundary condition. Is it possible to use both conditions together at the same region? If we have a cylindrical ...
1
vote
1answer
307 views

What is discontinuity in Vector Fields

I am reading David J. Griffiths and have a problem understanding the concept of discontinuity for E-field. The E-field has apparently to components. (How does he decompose the vector field into the ...
3
votes
2answers
181 views

The appearance of volume $V$ in the Fourier series representation of a periodic cubic system

In the textbook Understanding Molecular Simulation by Frenkel and Smit (Second Edition), the authors represent a function $f(\textbf{r})$ (which depends on the coordinates of a periodic system) as a ...
0
votes
2answers
731 views

Boundary conditions for static electric field

Consider a surface that carries surface charge density. In electrostatics, boundary conditions are studied by showing that there is a discontinuity in the normal component of the electric field across ...
1
vote
2answers
668 views

Image charges, laplace equation and uniqueness theorem

Consider a well-known problem of the electric field generated by a system composed of a point charge in proximity of a large earthed conductor. It is said that the potential due to an image charge ...