All Questions
29
questions
3
votes
3
answers
113
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 ...
- 220
1
vote
0
answers
171
views
Lienard-Wiechert Potential derivation in Wald's "Advanced Classical Electromagnetism" [closed]
I want to follow the Lienard-Wiechert potential derivation in Robert Wald's E-M book, page 179. I do not understand $dX(t_\text{ret})/dt$ on the right side. I assume the chain rule is applied and $x'^...
- 139
4
votes
1
answer
225
views
Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?
Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$
without ...
- 838
1
vote
1
answer
226
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
- 139
1
vote
1
answer
90
views
How to show the equivalence between Lagrangians?
I have a Lagrangian of a form:
$$\mathcal{L}=\frac{1}{2}\left (A_\mu g^{\mu\nu}\partial^2 A_\nu-A_\mu \partial^\mu \partial^\nu A_\nu\right ) $$
And I want to show that it is equivalent to the ...
- 47
0
votes
1
answer
60
views
How can I prove this relation between derivatives? [closed]
Consider coaixialcable with TEM. Nonstatic fields are being considered, i.e situation obeys $\nabla \times \mathbf {E}=-\frac{\partial \mathbf{B} }{\partial t} $
If I let a eletric field be described ...
- 1
-1
votes
1
answer
97
views
How we can prove this vector identity?
I was trying to rederive the formula of the angular momentum of electromagnetic field, and all the steps are clear for me except this one which I took from
"Photons and Atoms: Introduction to ...
- 11
4
votes
1
answer
108
views
What does $\mathbf{A}\cdot\nabla$ mean here?
What does $\mathbf{A}\cdot\nabla$ mean in an expression like $(\mathbf{A}\cdot\nabla)\mathbf B$?
I found this in Griffiths’ Classical Electrodynamics book and cannot figure it out.
0
votes
1
answer
1k
views
Commutator of covariant derivative and field $F_{\mu \nu}$
I am working with the covariant derivative and trying to show that the commutator of this derivative
$[D_\mu , D_\nu]$ is proportional to the field $F_{\mu \nu}$. That is, I need the final term to
be ...
user276724
3
votes
1
answer
435
views
Heaviside-Feynman formula derivation
I want to discuss derivation of Feynman-Heaviside formula.
The topic has already been discussed here but I can not put there any question that's why I'm making new post.
Deriving Heaviside-Feynman ...
0
votes
2
answers
593
views
How does a charged particle behave in a vector potential?
I know that a charged particle interacts with a magnetic field through the Lorentz force, thus knowing how it behaves in a given magnetic field.
However, I don't understand how a charged particle (be ...
0
votes
3
answers
169
views
Proof for Vector Identity
I am currently studying electrodynamic and came across the following vectoridentity, but I am unable to prove it:
$$ \vec{f} \times ( \nabla \times \vec{f} ) -\vec{f}(\nabla\cdot\vec{f}) = \nabla \...
- 3
2
votes
1
answer
213
views
Tensor Differentiation
In the book "Tensors, Relativity and Cosmology" the author derived Maxwell's Equation in covariant form using the EM field tensor Lagrangian $L=-\frac{1}{4}F^{jl}F_{jl}$ (source=0). One of the steps ...
- 1,047
0
votes
1
answer
415
views
Taylor expansion of scalar fields [closed]
Starting of with electrodynamics I have to compute the taylor expansion around $\vec{r} = 0$ of
$\psi (\vec{r}) = |\vec{r} - \vec{r_0}|^{\frac{3}{2}}$ where $\vec{r_0}$ is a constant vector up to ...
- 249
1
vote
1
answer
836
views
Derivation of curl of magnetic field [closed]
I am having trouble in one part of derivation of curl of magnetic field, from Biot-Savart law. The attached picture is from Griffiths - Introduction to Electrodynamics.
I got all the parts, but only ...
- 327
0
votes
2
answers
302
views
Canonical momentum of a 4-vector field
In a four-vector field theory,
we have a given Lagrangian:
$$\mathscr{L} = C_{1} (\partial_{\nu} A_{\mu}) (\partial^{\nu} A^{\mu}) + C_2 (\partial_{\nu} A_{\mu}) (\partial^{\mu} A^{\nu}) + C_3 A_{\mu} ...
- 112
1
vote
3
answers
143
views
Passing from curl to vector product
I don't understand how to obtain second equation with first part in the equation
$$
\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}.
$$
Can you ...
- 13
1
vote
1
answer
130
views
Divergence of a specific electrical field [closed]
I need to show that the divergence of the electrical field given as
$$\vec{E}=\vec{e_{\theta}}\frac{A\sin\theta}{r}\exp[i\omega(t-r/c)]$$
is zero. As the vector (in sperical coordinates) containes ...
- 11
-1
votes
1
answer
22k
views
Maximum electric field of a circular ring
How do you differentiate the equation for electric field of uniform ring
$$ E_x = \frac{kxQ}{(x^2+r^2)^{3/2}} $$to get the maximum at a point? My book says $x = r/\sqrt2$. I tried differentiating ...
- 307
0
votes
1
answer
58
views
Differential Operator
I am trying to understand the following expression
\begin{eqnarray}
e^{-ik.x}D_{\mu}D^{\mu}e^{ik.x} & = & e^{-ik.x}(i\partial_{\mu}+A_{\mu})(i\partial^{\mu}+A^{\mu})e^{ik.x}\\
& = & e^{...
- 53
1
vote
1
answer
285
views
Help with relativistic notation (Derivative of Lagrangian)
I am trying to learn QFT, but I haven't taken a course in general relativity so the relativistic notation stuff is taking me a bit to get used to. I do not understand how to do the following.
For a ...
- 1,001
1
vote
1
answer
70
views
Derive an equation related to magnetism [closed]
Solve the equations for $v_x$ and $v_y$ :
$$m\frac{d({v_x)}}{dt} = qv_yB \qquad m\frac{d{(v_y)}}{dt} = -qv_xB$$
by differentiating them with respect to time to obtain two equations of the form: $$...
- 119
2
votes
1
answer
2k
views
Derivatives with upper and lower indices
I'm studying classical and quantum field theory, but evaluating derivatives of fields (scalar and/or vector) described with upper and lower indices is somewhat new to me. I'm trying to evaluate
$$\...
- 187
2
votes
2
answers
4k
views
Total time derivative of magnetic vector potential $A$
I am looking at this document, which tries to establish the Lagrangian of the Lorentz force. Everything is fine, but I don't see why:
$$\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+\frac{\partial ...
- 594
0
votes
1
answer
105
views
I need help with divergence and gradient? [closed]
$$A_z = \mu{\frac{e^{-jBr}}{4\pi r}}∫I(z')e^{jBz'\cos\theta}dz'$$
Midway into my question, I want to compute:
$$-j\left( \frac{\nabla(\nabla\cdot A) }{w\mu\varepsilon} \right).$$
Symbols like $ w, \...
- 113
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 $- \vec a \frac{2}{r^3}$, ...
- 4,788
1
vote
2
answers
4k
views
Derivative of the magnetic field to the vector potential
So the magnetic field is defined with the vector potential A as:
$$\mathbf{B}=\nabla\times\mathbf{A}.$$
How would I calculate the derivative:
$$\frac{\delta}{\delta\mathbf{A}}|\mathbf{B}|^2$$
I ...
- 3,112
2
votes
3
answers
498
views
About field gradient
I read the term field gradient in most of the article about magnetic field. I search it online but most of the explanation is about the math. I wonder in physics, what the gradient field really mean? ...
- 2,383
4
votes
2
answers
2k
views
Derivatives of Dirac delta function and equation of continuity for a single charge
For a single charge $e$ with position vector $\textbf R$, the charge density $\rho$ and and current density $\textbf{j}$ are given by:
\begin{equation} \rho(\textbf{r},t)= e\,\delta^3(r-\textbf{R}(t))...
- 1,352