Linked Questions

4
votes
4answers
325 views

What's the commutator of $|\mathbf{\hat{x}}|$ and $|\mathbf{\hat{p}}|$? [closed]

Straight to the point: what's the result of the commutator of the magnitude of the position and the momentum operators and how can I approach it, i.e., $[|\mathbf{\hat{x}}|,|\mathbf{\hat{p}}|]=$ ? My ...
0
votes
0answers
23 views

Prove the given equation [duplicate]

While solving for the curl of the magnetic field($\vec \nabla\times \vec B = \mu_0 \vec J$), I got one formula which is written as $$\vec \nabla \cdot \frac{\hat r}{r^2} = 4\pi \delta^3 \vec r \tag{1}$...
0
votes
3answers
304 views

Verify that the electrostatic potential satisfies the Poisson equation [closed]

I'm reading Sect1.7 of Jackson's classical electrodynamics but I have trouble following his argument. Could someone help explain how exactly the Laplacian is evaluated in 1.30? Is it calculated with ...
3
votes
1answer
227 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 ...
0
votes
1answer
2k views

If the electrostatic Coulomb force followed inverse cube law, what's implication on Gauss' law?

If the electrostatic force between two charged particles varied according to the inverse of the cube of distance between them, how would it affect the Gauss Law? Well we know if a metallic sphere is ...
0
votes
1answer
788 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}{\...
37
votes
4answers
7k views

Why is this vector field curl-free?

The curl in cylindrical coordinates is defined: $$\nabla \times \vec{A}=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\hat {\...
1
vote
0answers
54 views

Dirac delta function equation intuition and proof [duplicate]

What is the intuition and where should I find proof of this equation (do not know what its name is). It is used to derive Gauss law from Newton equation. $${\nabla \cdot \Bigg ( \frac{\vec{s}}{|\vec{s}...
22
votes
4answers
56k 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. ...
4
votes
1answer
485 views

Distributions (e.g., Dirac Delta): confused and unhappy [closed]

I am sorry that the following set of questions is very fuzzy and ill informed. I am a trained mathematician and now studying an undergraduate theoretical physics course. We use distributions. I have ...
5
votes
3answers
2k views

The magnetic field of a magnetic monopole

Let us define the magnetic field $$\vec{B} = g\frac{\vec{r}}{r^3}$$ for some constant $g$. How can we show that the divergence of this field correspond to the charge distribution of a single magnetic ...
2
votes
1answer
542 views

Differentiation and delta function

Need help doing this simple differentiation. Consider 4 d Euclidean(or Minkowskian) spacetime. \begin{equation} \partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ? \end{equation} where $a_\mu$ is a constant ...
13
votes
4answers
16k 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 ...
5
votes
6answers
6k 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 ...