All Questions
37 questions
5
votes
4
answers
386
views
Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 1,723
1
vote
1
answer
71
views
Meaning of colon symbol $:$ in optics
When I was reading some early days nonlinear optics paper/textbooks (particularly between 1960-1985), I often see expressions such as:
$\chi^{(2)}:\textbf{E}\textbf{E}$
or
$\nabla\textbf{E}:\partial \...
- 78.1k
1
vote
1
answer
604
views
Visual representation of the curl of the magnetic vector potential!
I know that the electric field (a vector field) is the result of the gradient of the electric potential,which is a scalar field of the type: $\Phi$ : $\mathbb{R}^3 \rightarrow \mathbb{R}$. So the ...
CommunityBot
- 1
3
votes
1
answer
163
views
Laplace-Beltrami operator for a vector field
For a scalar field $\varphi$, the "wave" operator is defined as follows:
$$\Box \varphi \equiv g^{ab}\nabla_a\nabla_b~\varphi = \frac{1}{\sqrt{|g|}}\partial_a\left\{\sqrt{|g|}~g^{ab}~\...
- 8,900
3
votes
1
answer
310
views
What's the physical meaning of Curl of Curl of a Vector Field?
The curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$
Now, curl means how much a vector field rotates ...
- 4,609
5
votes
1
answer
5k
views
What's the physical meaning of vector Laplacian of Electric field intensity?
Could someone explain to me the physical meaning of vector Laplacian of Electric field intensity?
Where vector Laplacian means: $$\nabla^2 \mathbf{E} = \nabla(\nabla \cdot \mathbf{E}) - \nabla \times ...
- 213k
3
votes
3
answers
116
views
Finding the vector potential
$$\nabla\times\mathbf{B}=\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^2\mathbf{A}=\mu_0\mathbf{J}\tag{5.62}$$
Whenever I try to work this out and ...
- 1,830
0
votes
1
answer
183
views
Why the divernce of this magnetic field is not zero?
I am working on a project on which I need to calculate the geomagnetic field in different coordinates. When I use the conventional form of the dipole field in spherical coordinates:
$$\vec{B}_{r,\phi}=...
- 1,918
1
vote
3
answers
1k
views
Field strength tensor in spherical coordinates
I'm getting confused by the change of coordinates when calculating the electromagnetic tensor in spherical coordinates. In particular I know that in cartesian coordinates:
$$F_{\mu \nu}=\partial_{\mu}...
- 71.5k
0
votes
1
answer
47
views
Divergence applied to vector field, question
Divergence is defined as a scalar valued function:
$$
\left. \operatorname{div} \mathbf{F} \right|_\mathbf{x_0} = \lim_{V \to 0} \frac{1}{|V|} \int\int_{\scriptstyle S(V)}
\mathbf{F} \cdot \mathbf{\...
- 245
0
votes
1
answer
91
views
Poynting Theorem Derivation
I found this equation when I read about Poynting's theorem in Griffith's book.
$$
\textbf{B}\cdot\frac{\partial\textbf{B}}{\partial t}=\frac{1}{2}\frac{\partial}{\partial t}(B^2)
$$
Can anyone please ...
- 213k
0
votes
3
answers
114
views
What and how are you measuring with $\frac{dy}{dx}$, $dx$, $\mathbf{\nabla} \cdot$ and $\mathbf{\nabla} \times$?
In geometry, the gap between the mathematical model, be it axiomatic or algebraic, and the measurable real world is often non-existent. We have the intuition going from the mathematical model to world ...
- 25.4k
1
vote
1
answer
140
views
Intuitive Definition of Curl and Stokes' Theorem
I am asking this question on SE Physics because it's about an intuitive derivation, not a rigorously made proof of a theorem.
Reading Mathematical Methods for Physics and Engineering by Riley, et al. (...
- 78.1k
0
votes
1
answer
132
views
Vector calculus in Electromagnetism [closed]
I found a problem which had $$\partial_i (A_j \vec{G})= (\vec{\nabla} .\vec{ A} )\vec{G}+ (\vec{A}.\nabla) \vec{G} $$ but my problem is what does $$\partial_i (A_j \vec{B})$$ even mean? it doesn't ...
CommunityBot
- 1
-1
votes
1
answer
98
views
How we can prove this vector identity?
I was trying to rederive the formula of the angular momentum of electromagnetic field, and all the steps are clear for me except this one which I took from
"Photons and Atoms: Introduction to ...
- 4,314
4
votes
1
answer
111
views
What does $\mathbf{A}\cdot\nabla$ mean here?
What does $\mathbf{A}\cdot\nabla$ mean in an expression like $(\mathbf{A}\cdot\nabla)\mathbf B$?
I found this in Griffiths’ Classical Electrodynamics book and cannot figure it out.
- 52.2k
3
votes
1
answer
838
views
Vector and Scalar Helmholtz equation
This is closely related to this recent question
The vector Helmholtz equation is
\begin{align}
(\nabla^2 + k^2)\boldsymbol{u} = 0
\end{align}
The scalar Helmholtz equation is
\begin{align}
(\nabla^...
- 2,073
2
votes
2
answers
1k
views
Why do we need the curl and divergence on Maxwell equations? [closed]
Is there a particular reason to use curl and divergence on the description of electromagnetic fields? Given boundary conditions, if someone knows the curl and divergence of any field, is it always ...
- 213k
3
votes
2
answers
639
views
The strange character of operator $\nabla$
I was first introduced to the mathematical operation gradient, divergence and curl not in Mathematics but during my studies of Electromagnetism. As you all know learning Maths from a Physics teacher ...
- 293
0
votes
2
answers
609
views
How does a charged particle behave in a vector potential?
I know that a charged particle interacts with a magnetic field through the Lorentz force, thus knowing how it behaves in a given magnetic field.
However, I don't understand how a charged particle (be ...
- 10.1k
5
votes
4
answers
5k
views
Why are divergence and curl related to dot and cross product?
I've been reading Griffith's intro to electrodynamics and I've been a bit confused about his explanation of divergence and curl. I don't understand how divergence is the dot product of a gradient ...
1
vote
0
answers
31
views
Divergence of a vector which has explicit and implicit position dependence
I am doing EMT and I am trying to calculate the divergence of this current density given as,
$$\vec{J}(\vec{r}', t_r) = \vec{J}(\vec{r}', t - \frac{|\vec{r}-\vec{r}'|}{c})$$
for $\vec{r} = (x,y,z)$ ...
- 213k
2
votes
2
answers
822
views
How does the physical meaning of curl is in agreement with these scenarios?
In the foundation chapters of Electrodynamics I was introduced to concept of curl of a vector field. It was defined as follows $$ \nabla \times \mathbf A = \begin{vmatrix}
\hat{i} &...
CommunityBot
- 1
3
votes
1
answer
1k
views
Erratum in Griffith's Introduction to Electrodynamics
Applying the divergence to Eq. $47$, we obtain
$$ \mathbf{\nabla} \cdot \mathbf{B} = \frac{\mu_{0}}{4\pi} \int \nabla \cdot \left( \mathbf{J} \times \ \frac{\hat{\mathbf{r}}}{r^2}\right) d\tau^{'}. \...
CommunityBot
- 1
0
votes
2
answers
96
views
Why does $\vec{B}\cdot\frac{\partial \vec{B}}{\partial t}=\frac{1}{2}\frac{\partial}{\partial t} (B^2)$?
$$\vec{B}\cdot\frac{\partial \vec{B}}{\partial t}=\frac{1}{2}\frac{\partial}{\partial t} (B^2)$$
Griffiths states this result in his derivation of the Pontying vector, but I have absolutely no idea ...
- 213k
7
votes
6
answers
15k
views
Why is curl of current density $\nabla \times \vec{J}$ equal zero?
I am revisiting the derivation for $\nabla \cdot \vec{B} = 0$ in magnetostatics for the field $\vec{B}(\vec{r})$ of a charge $q$ at position $\vec{0}$ with velocity $\vec{v}$. It proceeds like
\begin{...
- 207
0
votes
3
answers
185
views
Proof for Vector Identity
I am currently studying electrodynamic and came across the following vectoridentity, but I am unable to prove it:
$$ \vec{f} \times ( \nabla \times \vec{f} ) -\vec{f}(\nabla\cdot\vec{f}) = \nabla \...
- 213k
1
vote
1
answer
130
views
Divergence of a specific electrical field [closed]
I need to show that the divergence of the electrical field given as
$$\vec{E}=\vec{e_{\theta}}\frac{A\sin\theta}{r}\exp[i\omega(t-r/c)]$$
is zero. As the vector (in sperical coordinates) containes ...
CommunityBot
- 1
1
vote
2
answers
350
views
Problem with the Landau gauge
I'm having a very simple problem which probably has an equally simple answer. I'm following the wikipedia article: https://en.wikipedia.org/wiki/Landau_quantization
We have a uniform magnetic field ...
- 859
1
vote
3
answers
143
views
Passing from curl to vector product
I don't understand how to obtain second equation with first part in the equation
$$
\nabla \times \vec A_0 e^{-j \vec k\cdot \vec r} = -j\vec k\times \vec A_0 e^{-j \vec k\cdot \vec r}.
$$
Can you ...
- 16.4k
1
vote
2
answers
2k
views
Explanation about Curl in electromagnetism [duplicate]
what does curl means in electromagnetism? How can we derive relation between curl and gradient in electromagnetism?
- 383
1
vote
1
answer
1k
views
Physical interpretation of curl of curl of Electric field vector
What is the physical interpretation of curl of curl of an E-field vector?
I know that this gives the expression for an EM wave using the Maxwell equations but, I want to understand what exactly curl ...
- 1,246
-1
votes
2
answers
383
views
Derivation of divergence
I was reading about the theory of electromagnetism and got stuck here:
I am getting confused with this Taylor series expansion. Though I know how to expand the series about a point $P$, but in the ...
- 241
0
votes
1
answer
77
views
Differentiating vector potentials
I am investigating a lagrangian with the end game of determining a Hamiltonian to describe a SUSY charged particle in an electric field however I am having a fundamental difficulty with my maths.
If ...
- 213k
0
votes
2
answers
1k
views
Divergence of vector potential [closed]
I was given the vector potential $$\vec A (\vec r) = - \vec a \times \nabla \frac{1}{r}$$ with a constant vector $\vec a$. Now, I found the $\vec B$ field which is I think $- \vec a \frac{2}{r^3}$, ...
- 719
5
votes
5
answers
7k
views
What is divergence?
What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
- 137k
1
vote
1
answer
253
views
Curl of a vector field with two different systems of coordinates
Let
$$\mathbf{H} = H_x \mathbf{u}_x + H_y \mathbf{u}_y + H_z \mathbf{u}_z$$
be a vector field whose components are defined with respect to the unit vectors $\mathbf{u}_x$, $\mathbf{u}_y$ and $\...
CommunityBot
- 1