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4
votes
4answers
148 views

Divergence of $\frac{ \hat {\bf r}}{r^2}$ , what is the 'paradox'?

I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2}$ , now from the book, Griffiths says: Now what is the paradox, exactly? ...
1
vote
2answers
144 views

Mathematical rigorous definition for an electrical dipole

I've been reading Laurent Schwartz's Mathematics for the physical sciences, and in the chapter on distributions he makes many cool examples of ways to define in a mathematical rigorous way certain ...
0
votes
2answers
44 views

Is it true that any electrostatic charge density can be represented by Dirac Delta function? Give me a general example

My college professor said that any charge density can be replaced by Dirac Delta function but how I want to know.
0
votes
1answer
27 views

What is the volume charge density (in spherical coordinates)?

What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $R$ and total charge $Q$, centered at the origin? Give your answer in terms ...
0
votes
1answer
45 views

What is dipolar charge distribution?

An electric dipole is a system of two opposite point charges when their separation goes to zero and their charge goes to infinity in a way that the product of the charge and the separation remains ...
1
vote
1answer
66 views

Integrating Laplace's equation over a sphere

The Wikipedia page on Laplace's equation states that if the Laplacian of $u$ is integrated over any volume that encloses the source point, $$\iiint_V \nabla \cdot \nabla u \, d^3V =-1.$$ I can'...
1
vote
2answers
79 views

How can we argue $\nabla\cdot v = 0$ when ${\mathscr r} \ne 0$ on this vector function?

I am dealing with the vector field: $$v = \dfrac{\hat{\mathscr r} }{{\mathscr r}^2}$$ And I am studying its divergence. If we compute it we get: $$\nabla\cdot\left(\dfrac{\hat{\mathscr r} }{{\...
1
vote
0answers
53 views

Why Gauss divergence theorem isn't working? [duplicate]

$\vec{E}$ is electric field $r$ is distance between source and field points $\hat{r}$ is a unit vector from source point to field point $x,y,z$ are Cartesian coordinates of field point ...
0
votes
2answers
84 views

How to plot the graph of this expression which involves Dirac delta function?

I was doing a problem on electrostatics which required finding the charge density from the given electric field and then plot a graph of the charge density. I was able to find the charge density which ...
1
vote
2answers
101 views

Can $E=\frac{q}{4\pi\epsilon_0 r^2}$ be directly derived from differential form of Maxwell equations?

The electric field of a point charge $q$ is well known to be $$\mathbf E=\frac{q}{4\pi\epsilon_0 |\mathbf r|^3}\hat{\mathbf r}$$ This can be derived easily from integral form of Gauss’s law. Taking $...
3
votes
1answer
111 views

How to derive the $\frac{4\pi}{3}\vec{p}\delta^3(\vec{r})$ element for the dipole field, from its potential?

This might be a bit more general question about how to figure out what is the appropriate (delta) expression in singular points, but e.g. for the dipole, we can derive its potential by a taylor ...
3
votes
3answers
2k views

Divergence of Electric Field Due to a Point Charge

I am trying to formally learn electrodynamics on my own (I only took an introductory course). I have come across the differential form of Gauss's Law. $$ \nabla \cdot \mathbf E = \frac {\rho}{\...
1
vote
1answer
79 views

Is the static electric field $\vec{E}$ defined and finite where charges are located?

Suppose you have a continuous charge distribution, e.g. a wire, a disk, or a plane, represented by a subset $D\subset \mathbb{R^3}$. The charge density is $\rho: \mathbb{R^3}\rightarrow \mathbb{R}$. ...
1
vote
1answer
215 views

The charge density $\rho_{\infty}$ of the sphere

I have two charged hemispheres (which are very close to each other, we consider them now as a sphere) with the charge density given as $\rho = \frac {Q}{\pi a^3 \frac {4}{3+n}} (\frac{r}{a})^n$, ...
4
votes
3answers
143 views

Obtaining the charge from the charge density using distribution theory

In electrostatics, for several reasons, it seems that the correct way to understand the charge density $\rho$ isn't as a function $\rho : \mathbb{R}^3\to \mathbb{R}$, but rather as a distribution $\...
0
votes
1answer
326 views

Differentiating the electric field in Gauss's law, I get zero charge density. Can anyone help me where am I going wrong?

$$\mathbf E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2} \mathbf e_r$$ $$\nabla \cdot\mathbf E = \frac{\rho_V}{\epsilon}$$ \begin{align} \implies \nabla\cdot\mathbf E & = \frac{1}{r^2}\frac{\partial}{\...
-3
votes
1answer
192 views

Why does the finite difference script for solving Poisson equation not work for delta function charge? How to fix it?

I know the charge density at the plates of a capacitor, from which the applied voltage and potential profile is to be calculated. Using the finite difference method in MATLAB script I solved the ...
0
votes
1answer
332 views

Problem with electric potential involving a delta function [closed]

Consider $V(r)=\dfrac{Ae^{-\lambda r}}{r}$ with $\lambda, A$ constants with appropriate units. Calculate the electric field, the charge density $\rho$ and the total charge $Q_{tot}$. (HINT: In a ...
0
votes
1answer
264 views

Dirac Delta Function

In the image above the second faint part is the answer to the question c). The answer starts with the word evidently, and I am confused why it is evident. Also, since we are considering a shell, ...
1
vote
3answers
5k views

Dirac delta function and volume charge density

I just got introduced to the Dirac delta function and one of the questions was to express volume charge density $\rho({\bf r})$ of a point charge $q$ at origin. I saw that the answer is related to ...
1
vote
1answer
188 views

Using Delta Dirac function as a mathematical tool in Green's functions

So, I was studying green's functions and in general I understood that if I have an operator $\mathscr{O}$ that acts of a function $h_1(\vec{r})$ such that $$\mathscr{O}h_1(\vec{r})=h_2(\vec{r})$$ ...
1
vote
1answer
1k views

Charge density as Dirac delta function

I little bit confused while I'm trying to convert a cylindrical charge distribution to spherical. The question is: A uniformly charged thin disk of radius $a$ and surface charge density $\sigma$ ...
1
vote
1answer
2k views

Surface density charge, divergence of the electric field and gauss law

It´s known that the divergence of the electric field at a certain point is given by this formula: $$\nabla \cdot E=\dfrac{\rho (r)}{\epsilon_{0}}$$ Being $\rho (r)$ the volume charge density at that ...
3
votes
3answers
988 views

What does the Dirac delta function physically do while deriving Gauss Law form Coulomb's law?

While doing this derivation, the the source coordinates are mentioned as "$s$" and the coordinate of the point at which field is to be calculated is mentioned as "$r$". Kindly follow this Wikipedia ...
1
vote
1answer
395 views

Why is it true that Laplace's equation does not hold within the sphere in this case?

Find the average potential over a spherical surface of radius $R$ due to a point charge $q$ located inside. (In this case Laplace's equation does not hold within the sphere) This is a question from ...
2
votes
2answers
2k views

Definition of a line charge with Dirac delta function [closed]

Is the following statement correct for a line charge distribution $λ(x)$? $$ρ(\mathbf r)=λ(x)δ(y)δ(z)$$ If yes - what does it say?
4
votes
2answers
3k views

Gauss' law in differential form for a point charge

I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that $ \nabla\cdot E=\frac{1}{\epsilon 0}\rho$, but when ...
2
votes
1answer
168 views

How do you know when you need to use distributions to represent charge densities? [closed]

I tried to solve a problem using Gauss' law in the following way. Let's assume we have a spherical shell of radius $R$ with a charge $Q$ being homogenously distributed on its surface. I am trying to ...
0
votes
1answer
113 views

Divergence of a vector field, going through the math [closed]

The example I'm working on has this given identity: $\bigtriangledown \cdot \mathbf{\bar{r}}=3$. The question is: find the divergence of a vector field $\bar{\mathbf{E}}=\frac{\mathbf{r}}{r^{3}}$. ...
0
votes
1answer
594 views

Green function solutions in electrostatics

I have a conducting plate on $x$-$y$ plane. So I have a boundary condition at $z=0$ $\Phi=0$ but, for $z>0$ I have a point charge at z=a which is expected to create a potential. $$ \nabla^2\Phi=\...
2
votes
1answer
340 views

Is it equivalent to derive Gauss's law from discrete and continuous source distributions?

I've seen two derivations for Gauss's law in electrostatics. The first assumes a discrete charge distribution, the second a continuous one: Use superposition $$\vec{E}=\sum_{i=1}^n\vec{E}_i,$$ so ...
16
votes
4answers
36k views

Divergence of a field and its interpretation

The divergence of an electric field due to a point charge (according to Coulomb's law) is zero. In literature the divergence of a field indicates presence/absence of a sink/source for the field. ...
1
vote
2answers
267 views

Divergence of conservative electric field

I have a little doubt about the following: according Gauss law in the form of Maxwell's equation, we know that: $$ {\rm div} (D)~=~ \rho(v) $$ This just tells us that the electric field has nonzero ...
6
votes
4answers
11k views

Delta Dirac Charge Density question

I have to write an expression for the charge density $\rho(\vec{r})$ of a point charge $q$ at $\vec{r}^{\prime}$, ensuring that the volume integral equals $q$. The only place any charge exists is at $...
11
votes
4answers
10k views

How is Gauss' Law (integral form) arrived at from Coulomb's Law, and how is the differential form arrived at from that?

On a similar note: when using Gauss' Law, do you even begin with Coulomb's law, or does one take it as given that flux is the surface integral of the Electric field in the direction of the normal to ...
24
votes
2answers
18k views

What are the units or dimensions of the Dirac delta function?

In three dimensions, the Dirac delta function $\delta^3 (\textbf{r}) = \delta(x) \delta(y) \delta(z)$ is defined by the volume integral: $$\int_{\text{all space}} \delta^3 (\textbf{r}) \, dV = \int_{-...
5
votes
6answers
5k views

What is the origin of the Dirac delta term in the dipole electric field?

I am a bit lost how one has deduced the formula for electric field with electric dipole because of some inconsistency between different sources. The Wikipedia article contains a delta function in the ...
12
votes
5answers
4k views

Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is $$\nabla^2\frac{q}{r}~...