All Questions
Tagged with differentiation electromagnetism
109 questions
15
votes
5
answers
2k
views
What does it mean for a physical quantity if its mixed second partial derivatives are not equal?
This goes for every problem (either in electromagnetism or fluid dynamics) that has to do with vector fields. Say we have a fluid flowing in a closed circular pipe (or an electromagnetic field, the ...
- 7,860
11
votes
2
answers
577
views
Is $∂_\mu + i e A_\mu$ a "covariant derivative" in the differential geometry sense?
I have heard the expression "$∂_\mu + i e A_\mu$" referred to as a "covariant derivative" in the context of quantum field theory. But in differential geometry, covariant derivatives have an ostensibly ...
- 3,566
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
6
votes
4
answers
3k
views
What is the current of a capacitor when the derivative of voltage is undefined?
This is from the textbook I am reading:
I know this equation for capacitors:
$$i=C\cdot \frac { dv }{ dt }$$
Here is my question: how can diagram (a) be allowed if the derivative of the voltage ...
- 63
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):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 61
5
votes
5
answers
7k
views
What is divergence?
What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
- 1,660
5
votes
4
answers
5k
views
Why are divergence and curl related to dot and cross product?
I've been reading Griffith's intro to electrodynamics and I've been a bit confused about his explanation of divergence and curl. I don't understand how divergence is the dot product of a gradient ...
5
votes
1
answer
5k
views
What's the physical meaning of vector Laplacian of Electric field intensity?
Could someone explain to me the physical meaning of vector Laplacian of Electric field intensity?
Where vector Laplacian means: $$\nabla^2 \mathbf{E} = \nabla(\nabla \cdot \mathbf{E}) - \nabla \times ...
- 51
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))...
- 1,412
4
votes
1
answer
230
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 ...
- 862
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.
4
votes
2
answers
157
views
Local symmetry restoration via a gauge field
In the book Quantum Field Theory for the Gifted Amateur, the author stated that, having a field that transforms locally via $\psi(x) \rightarrow \psi(x)e^{i \alpha(x)}$ will destroy local symmetry -...
- 165
4
votes
2
answers
59
views
Where does one more '$\rm m$' come from in the units?
$$\nabla \times A = B$$
$A$ is vector magnetic potential, $\mathrm{Wb/m}$
$B$ is magnetic field intensity, $\mathrm{Wb/m^2}$
Where does one more m come from for $B$? Is that from the gradient operator ...
- 285
4
votes
1
answer
3k
views
Is the differential form of Faraday-Henry equation ( Curl(E)= - dB/dt) always valid?
My textbook suggests that the integral form of the law is evident from experiments, while the differential form can be obtained by considering a closed curve, constant in time, so that it is ...
4
votes
2
answers
271
views
Conventions regarding partial derivatives
Look at this expression:
$$\frac{\partial}{\partial t} (V-\mathbf{v}\cdot\mathbf{A}).$$
This expression occurs in Griffiths EM book (4th ed, p.444). $V=V(\mathbf{r},t)$is the scalar potential, $\...
- 1,417
3
votes
2
answers
739
views
Derivation of curl of magnetic field in Griffiths
Can someone please derive how $$\frac{d}{dx} f(x-x') = -\frac{d}{dx'} f(x-x')~?$$
In Griffiths electrodynamics, this is directly mentioned. I'm really confused, can someone elaborate!
- 453
3
votes
3
answers
579
views
How does Kirchhoff's voltage law relate to the spatial derivative of voltage?
I'm reading this libretexts article on the basics of transmission line theory. In it, they include this circuit diagram as a model of a uniform transmission line:
They then say that applying ...
- 2,373
3
votes
1
answer
1k
views
Erratum in Griffith's Introduction to Electrodynamics
Applying the divergence to Eq. $47$, we obtain
$$ \mathbf{\nabla} \cdot \mathbf{B} = \frac{\mu_{0}}{4\pi} \int \nabla \cdot \left( \mathbf{J} \times \ \frac{\hat{\mathbf{r}}}{r^2}\right) d\tau^{'}. \...
user240696
3
votes
2
answers
639
views
The strange character of operator $\nabla$
I was first introduced to the mathematical operation gradient, divergence and curl not in Mathematics but during my studies of Electromagnetism. As you all know learning Maths from a Physics teacher ...
user240696
3
votes
3
answers
116
views
Finding the vector potential
$$\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}$$
Whenever I try to work this out and ...
- 220
3
votes
1
answer
92
views
Bianchi identity in EMT [closed]
$ ∇_a∇_b F_{ab} = 0 $ ($F_{ab}$ Faraday tensor in EMT.)
proof is given by
"To see this, assume a Minkowski spacetime for simplicity and adopt
Cartesian coordinates, so that the covariant ...
- 81
3
votes
1
answer
454
views
Heaviside-Feynman formula derivation
I want to discuss derivation of Feynman-Heaviside formula.
The topic has already been discussed here but I can not put there any question that's why I'm making new post.
Deriving Heaviside-Feynman ...
3
votes
1
answer
513
views
Gradient in the Frenet-Serret coordinate
I was simply thinking that the gradient in the Frenet-Serret coordinate at a particular point is similar to the gradient in the Cartesian coordinate. I simply assumed that Frenet space is an ...
- 119
3
votes
1
answer
310
views
What's the physical meaning of Curl of Curl of a Vector Field?
The curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$
Now, curl means how much a vector field rotates ...
- 121
3
votes
1
answer
838
views
Vector and Scalar Helmholtz equation
This is closely related to this recent question
The vector Helmholtz equation is
\begin{align}
(\nabla^2 + k^2)\boldsymbol{u} = 0
\end{align}
The scalar Helmholtz equation is
\begin{align}
(\nabla^...
- 15.2k
3
votes
2
answers
164
views
Differentiability of electric field due to bounded volume charge distribution
In books on electromagnetism, one often sees expressions of Maxwell's equations like $\nabla \cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$. These expressions make sense if $\mathbf{E}$ (which is ...
- 802
3
votes
1
answer
67
views
"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
- 283
3
votes
1
answer
163
views
Laplace-Beltrami operator for a vector field
For a scalar field $\varphi$, the "wave" operator is defined as follows:
$$\Box \varphi \equiv g^{ab}\nabla_a\nabla_b~\varphi = \frac{1}{\sqrt{|g|}}\partial_a\left\{\sqrt{|g|}~g^{ab}~\...
- 603
3
votes
4
answers
638
views
Derivation of covariant derivative
I'm currently doing Introductory QFT and was confused about the origin of the additional terms in the covariant derivate. My understanding is as follows:
If we begin with the Dirac Lagrangian ...
- 100
3
votes
0
answers
98
views
Why do we assume electromagnetic fields to be doubly differentiable? [duplicate]
It seems like the identities of curl of gradient, divergence of curl, and the simple derivations of electromagnetic waves from Maxwell equations all rely on the symmetry (interchangeability of their ...
- 1,169
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 $$ \nabla \times \mathbf A = \begin{vmatrix}
\hat{i} &...
user240696
2
votes
2
answers
1k
views
Why do we need the curl and divergence on Maxwell equations? [closed]
Is there a particular reason to use curl and divergence on the description of electromagnetic fields? Given boundary conditions, if someone knows the curl and divergence of any field, is it always ...
- 43
2
votes
1
answer
45
views
Why does linear media produce a $1/2$ factor when using partial derivatives?
Oftentimes in Jackson's text there are certain remarks made about linear media (see pg. 226, ed. 2, for example) and there is often a simplification of partial differentials or variations made. For ...
- 134
2
votes
1
answer
2k
views
Derivatives with upper and lower indices
I'm studying classical and quantum field theory, but evaluating derivatives of fields (scalar and/or vector) described with upper and lower indices is somewhat new to me. I'm trying to evaluate
$$\...
- 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:
$$\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+\frac{\partial ...
- 604
2
votes
1
answer
280
views
Factor before Dirac delta in magnetic dipole field formula
I bumped into this formula for the magnetic induction field generated by a dipole, containing Dirac's delta, while studying hyperfine splitting: $$\textbf{B}(\textbf{r}) = \frac{2}{3}\mu_0 \textbf{m}\...
2
votes
1
answer
373
views
Gauge covariant derivative, - how do I get the field?
Suppose I wish to create a gauge covariant derivative from
$$
\psi(x)\to e^{ia(x)}\psi(x)
$$
I first note that the usual derivate is not covariant:
$$
\partial_x(e^{ia(x)}\psi(x))=i(\partial _xa(x))e^{...
- 1,558
2
votes
1
answer
215
views
Tensor Differentiation
In the book "Tensors, Relativity and Cosmology" the author derived Maxwell's Equation in covariant form using the EM field tensor Lagrangian $L=-\frac{1}{4}F^{jl}F_{jl}$ (source=0). One of the steps ...
- 1,047
2
votes
2
answers
598
views
4-Gradient vector and the Field strength tensor
Need some help evaluating the following 4-gradient, that of the gradient of the field strength tensor
$$F^{\mu\nu}=
\begin{bmatrix}
0 & -E_x & -E_y & -E_z\\\
E_x & 0 & -...
- 153
2
votes
1
answer
551
views
Equating $\partial_{t_r}$ = $\partial_{t}$ in the retarded potentials?
Im reading Griffiths E/M (4th edition) and came across something I don't understand: Page 446, footnote 8 which reads:
"Note that $\frac{\partial }{\partial t_r} = \frac{\partial }{\partial t}$ since ...
- 165
2
votes
3
answers
504
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
1
vote
3
answers
824
views
Why is emf equal to the rate of change of magnetic flux?
I don't understand how Faraday figured out that the emf induced when a magnet is moved in a coil would be equal to the rate of change of magnetic flux. Yes, I have that formula drilled in my head like ...
- 753
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:
$$F_{\mu \nu}=\partial_{\mu}...
- 771
1
vote
2
answers
812
views
Reason why dot notation isn't used for time derivatives in Maxwell's equations [closed]
Maxwell's equations seem to be usually written:
\begin{align}
\nabla \cdot \mathbf{E} &= \rho/\epsilon_0,\\
\nabla \cdot \mathbf{B} &= 0,\\
\nabla \times \mathbf{E} &= -\frac{\partial \...
user113773
1
vote
2
answers
4k
views
Derivative of the magnetic field to the vector potential
So the magnetic field is defined with the vector potential A as:
$$\mathbf{B}=\nabla\times\mathbf{A}.$$
How would I calculate the derivative:
$$\frac{\delta}{\delta\mathbf{A}}|\mathbf{B}|^2$$
I ...
- 3,132
1
vote
1
answer
71
views
Meaning of colon symbol $:$ in optics
When I was reading some early days nonlinear optics paper/textbooks (particularly between 1960-1985), I often see expressions such as:
$\chi^{(2)}:\textbf{E}\textbf{E}$
or
$\nabla\textbf{E}:\partial \...
- 173
1
vote
1
answer
258
views
Derive interaction lagrangian for KG equation in QED
The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$
By ...
- 862
1
vote
1
answer
404
views
Difference between covariant derivatives in general relativity and electromagnetism
There are many similarities between the gauging of a $U(1)$ symmetry to obtain the physics of electromagnetism and the gauging of diffeomorphisms to obtain the physics of general relativity. In ...
1
vote
3
answers
143
views
Passing from curl to vector product
I don't understand how to obtain second equation with first part in the equation
$$
\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}.
$$
Can you ...
- 13
1
vote
1
answer
253
views
Curl of a vector field with two different systems of coordinates
Let
$$\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 $\...
- 777