All Questions
Tagged with electromagnetism mathematical-physics
71
questions
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16
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Effects of Localized Medium Changes on Field Propagation
I've studied various theories related to fields. These theories often include equations describing how the activity of a source is transmitted to other locations. The properties of the medium ...
- 695
2
votes
2
answers
158
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Non-orientability in electromagnetism
I'm currently studying E&M and I have a question related to the mathematical formalism of the theory. Electrodynamics depends heavily on the divergence and Stokes's theorem which in their ...
- 37
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0
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25
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Charge Density Of A Conducting Strip
I am currently doing a project which requires me to figure out the charge density of a strip. Assume that the strip is isolated in a vacuum.
Assume the strip is 1 dimensional, kind of like a rod. What ...
1
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0
answers
37
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Hertzian Dipole: Why is there no longer a phase shift at $\frac\lambda2$? [closed]
Today we learned about the Hertzian Dipole. Out teacher told us that the length of the wire connecting the two capacitor plates is $l=\frac\lambda2$. He also stated that there is a no more phase shift ...
- 11
2
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0
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101
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Free electromagnetic field BV action
I am trying to write down the extended BV-action of the free electromagnetic field in a physicist notation, but I don't find it anywhere. I found the following formula in example 3.1. of the paper ...
- 689
3
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3
answers
571
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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 ...
- 2,327
1
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0
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118
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Prandtl boundary layer equations for two-dimensional steady laminar flow of incompressible fluid over a semi-infinite plate are given by [closed]
Prandtl boundary layer equations for two-dimensional steady laminar flow of incompressible fluid over a semi-infinite plate are given by
jpg
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1
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0
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40
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Mathematical equivalent of Fundamental nature of charge [closed]
How to mathematically represent the fact that electric charge is a fundamental quantity? i.e. that it cannot be explained in terms of other things, for example, the normal force can be explained as ...
0
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1
answer
213
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Classical Green function
What is the physical reason why the classical Green's function is not defined as a principle value integral?
In a recent discussion (Classical Green's function) it was said that the classical ...
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1
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1
answer
109
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Completeness of Landau basis
We know that the Landau Hamiltonian (uniform magnetic field) is diagonalized by wavefunctions $|n,m\rangle,n,m\in \mathbb{N}$ in the symmetric gauge. However, does this set of functions form a "...
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1
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85
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2 magnet bars are placed in same plane and one of them can rotate freely. The relation as system is balanced [closed]
Each magnet is in same plane.
$$ \ell \ll r $$
The magnet1 has been fixed.
The magnet2 can rotate with center of the magnet itself.
$$ \theta_{1} ~~,~~ \theta_{2} :=\text{each angle between the ...
6
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5
answers
705
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How are vector quantities in three dimensions (velocity, electric field, etc.) modeled in mathematical physics?
In introductory courses, vectors are defined as objects with direction and magnitude. I guess everyone has arrows in mind when talking about vectors and that's probably the most intuitive description, ...
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1
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1
answer
30
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Permissible Electrostatic Potential
Let us consider a $1D$ real function $V(x)$. When is this a classical electrostatic potential?
My take on the problem:
$V(x)$ must be differentiable everywhere. In fact, we should be able to ...
- 785
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votes
2
answers
111
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Linking the de Rham bundle/complex over spacetime to the gauge bundle
In some textbooks, the Maxwell equations are stated in a very simple mathematical form (up to multiplicative constants coming from the system of units being used):
$$ \begin{array} \mbox{d}F =0, \\ \...
- 4,361
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23
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Source for Learning? [duplicate]
I am an very ameteur mathematician and physicst (If I can say mathematician and physicst to myself xD). I want to learn topics in physics. Like electromagnetism, mechanic, thermodinamics etc. But I ...
2
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1
answer
235
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How to find convergence of conditionally convergent series obtained while calculating the electrostatic potential energy of a NaCl crystal?
I was reading Electricity and Magnetism by E M Purcell and there in the first chapter there is an attempt to estimate the electrostatic potential energy of the crystal lattice of a NaCl crystal.
...
- 1,147
3
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1
answer
455
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Why are Cauchy boundary conditions an over-specification of boundary conditions for solving Poisson’s equation?
I was referred to Physics.SE by the following content published in Jackson’s Classical Electrodynamics:
This rather surprising result [the fact that the potential within a charge-free volume is ...
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1
answer
3k
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The use of Helmholtz decomposition
Examining the article on Wikipedia Helmholtz decomposition, compatible with the explanations of the book Introduction to Electrodynamics $4^{\mathrm{th}}$ edition David J. Griffiths §1.6 the theory of ...
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1
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0
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47
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Why shouldn't I choose my boundary limits corresponding to the direction I'm integrating?
I have a question regarding the choice of boundary limits when it comes to vector integrals. Why shouldn't I always choose the boundary limits corresponding to the direction I'm integrating. I.e why ...
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2
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3
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427
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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 ...
7
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2
answers
1k
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Facing a paradox: Earnshaw's theorem in one dimension
Consider a one-dimensional situation on a straight line (say, $x$-axis). Let a charge of magnitude $q$ be located at $x=x_0$, the potential satisfies the Poisson's equation $$\frac{d^2V}{dx^2}=-\frac{\...
- 26.8k
3
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1
answer
294
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Formal Connection Between Symmetry and Gauss's Law
In the standard undergraduate treatment of E&M, Gauss's Law is loosely stated as "the electric flux through a closed surface is proportional to the enclosed charge". Equivalently, in differential ...
- 130
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0
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331
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Wave Equations from Decoupling Maxwell's Equations in Bianisotropic Media
For several days now, I have been trying to decouple Maxwell's equations in bianisotropic media so that I end up with a form that involves only one variable (of E, D, B, H), i.e. a so-called 'wave ...
- 591
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3
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559
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What are good non-paraxial gaussian-beam-like solutions of the Helmholtz equation?
I am playing around with some optics manipulations and I am looking for beams of light which are roughly gaussian in nature but which go beyond the paraxial regime and which include non-paraxial ...
- 134k
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0
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286
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General proof of independence of TM and TE modes in a waveguide
In electromagnetic field analysis for a typical waveguide that has a uniform cross section along its axial direction (say $z$), we often describe the E and H fields conveniently in terms of their ...
- 632
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1
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56
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For closed circuits, why can't we have more than one $f(r)$?
Force between current elements depends on a function of angles [$f(\eta, \theta, \theta^{\prime})$] and also on a function of distance between them [$f(r)$] .
For closed circuits, there are more ...
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2
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1
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107
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How are these multipole moments related to the ones from electrodynamics?
Let $f : \mathbb{R}^3\to \mathbb{R}$ be a continuous function of compact support. Its Fourier transform is
$$\mathfrak{F}[f](k)=\int f(x)e^{ikx}dx=\int f(x)\sum_{n=0}^\infty \dfrac{i^n}{n!}k_{a_1}\...
- 36.4k
4
votes
1
answer
577
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Can one force the octupole moments of a charge distribution (neutral and with vanishing dipole moment) to vanish using a suitable translation?
In a previous question, I noted that if you have a charge distribution with nonzero charge, then it is possible to choose an origin (at the centre of charge) which makes its dipole moment vanish, and ...
- 134k
3
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1
answer
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The electric field in phasor/complex notation
The electric field in phasor notation is often written
\begin{align}
\mathbf{E}(x,y,z,t)&=\Re\{\mathbf{E}_0\mathrm{e}^{j\phi}\mathrm{e}^{j\omega t}\}\\
&=\Re\{\tilde{\mathbf{E}}\mathrm{e}^{j\...
- 433
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3
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353
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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 $\...
- 36.4k
5
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1
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596
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Different possible solutions for the wave equation?
The Wave equation is:
$$\nabla^2\psi(\mathbf{x},t)-\frac{1}{c}\frac{\partial^2 \psi(\mathbf{x},t)}{\partial t^2}=f(\mathbf{x},t)$$
The Green function is then
$$\nabla^2G(\mathbf{x},t)-\frac{1}{c}\...
- 937
2
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1
answer
213
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Physical accuracy of Hankel function solution to cylindrical voltage wave propagation?
The well-known solution to an outward-travelling wave in cylindrical coordinates (in an unbounded medium) is the Hankel function of the first kind:
$$H^{\left(1\right)}_n (\rho,t) = \left(J_n (\rho,t)...
- 123
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1
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629
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Transformer universal EMF equation derivation
At this Wikipedia page, we've that the 'Transformer universal EMF equation' looks like:
$$\text{E}_{\text{rms}}=\frac{2\pi\times\text{f}\times\text{n}\times\text{a}\times\text{B}_{\text{peak}}}{\sqrt{...
- 291
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0
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279
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Is mathematical rigour irrelevant in most physics fields? [duplicate]
Are mathematical notions like closed sets, limits of sequences, measures, and function spaces basically irrelevant in the day to day work of a physicist? Naturally, such concepts are the foundations ...
- 659
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2
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583
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Calculating the potential on a surface from the potential on another surface
The question is short: If a charge (or mass) distribution $\rho$ is enclosed by surface $S_1$, I can calculate the electrostatic (or gravitational) potential on that surface by integrating $\rho(r') \ ...
- 6,273
6
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2
answers
2k
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Proof of Ampère's law from the Biot-Savart law for tridimensional current distributions
Let us assume the validity of the Biot-Savart law for a tridimensional distribution of current:$$\mathbf{B}(\mathbf{x})=\frac{\mu_0}{4\pi}\int_V \mathbf{J}(\mathbf{y})\times\frac{\mathbf{x}-\mathbf{y}}...
1
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1
answer
237
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response function and Fourier transform
A response function defined as the kernel of the following integral:
$\rho(t) = \int_{-\infty}^t \chi(t,t') E(t')dt'$ (1), where $\chi(t,t')$ is the response function.
Physically, it relates ...
- 171
3
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0
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966
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Mathematics of Surface Divergence and Surface Curl
While studying electrodynamics I found two functions - Surface Divergence and Surface Curl - that seem to condense the formulas for superficial discontinuities of the electric and magnetic fields ...
- 88
2
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1
answer
97
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Time dependent electric field: Mathematical expansion for local electric field
In many articles and books I see that local electric field is expanded as
$$\vec E_0(\vec r(t)) = \vec E_0(\vec R_0) − (\vec a(t) \cdot \nabla) \vec E_0(\vec R_0) \cos(\Omega t) + \ldots $$
For ...
- 1,593
2
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1
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166
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Are there Non-conformal maps encountered in Physics?
We always encounter Conformal maps in Physics, may be they are easier to study, but are there Non-Conformal transformations encountered in Physics anywhere?
if they are encountered, where are they ...
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1
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497
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Is EM interpreted in a principal or vector bundle?
I've read in a few places that EM is a $U(1)$-principal bundle; but is this correct? Isn't it rather an associated vector bundle using the adjoint representation of $U(1)$?
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0
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203
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Regarding Ampere's Circuital Law
If I am to show that Ampere's Circuital law holds true for any arbitrary closed loop in a plane normal to the straight wire, with its validity already established for the closed loop being a circle of ...
3
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1
answer
537
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Divergence Theorem, mathematical approach to Gauss's Law?
Let $D$ be a compact region in $\mathbb{R}^3$ with a smooth boundary $S$. Assume $0 \in \text{Int}(D)$. If an electric charge of magnitude $q$ is placed at $0$, the resulting force field is $q\vec{r}/...
user81012
5
votes
2
answers
417
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Path dependent phase in quantum mechanics
In elementary treatments of quantum mechanics, we are taught that the wavefunction of a single particle is complex valued ($\Psi : \mathbb{R}^3 \to \mathbb{C}$). In particular, the wavefunction has a ...
- 281
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2
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407
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Do there exist functions $\phi$ and $A$ such that $\vec E$ satisfies the Helmholtz Theorem $\vec E = -\nabla \phi + \nabla \times \vec A$?
Helmholtz Decomposition theorem stats:
"Let $\vec F$ be a vector field on a bounded domain $V$ in $\mathbb R^3$, which is twice continuously differentiable, and let $S$ be the surface that encloses ...
- 1,734
16
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2
answers
2k
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Why isn't the path integral defined for non-homotopic paths?
Context
In the Aharonov Bohm effect, there is a solenoid which creates a magnetic field. Since the electron cannot be inside the solenoid, the configuration space is not simply connected.
Question
I'...
- 12.4k
28
votes
1
answer
1k
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Electric charges on compact four-manifolds
Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form $\mathbb{R} \times M_3$ where the first factor ...
- 401
1
vote
3
answers
128
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What is the relationship between $V(t)$ and $V(x,y,z)$
I was recently asked this by a friend.
He told me that coming from a physics background, he does not understand $V(t)$ and he believes it is purely theoretical construct made up by circuit theorists....
- 1,734
0
votes
1
answer
108
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Charge and current density fields
The charge and current density fields in classical electromagnetism are scalar real number fields on space time manifold. But these fields diverge/become infinite in case of point charges, how is this ...
- 1,578
15
votes
4
answers
2k
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Electromagnetic field and continuous and differentiable vector fields
We have notions of derivative for a continuous and differentiable vector fields. The operations like curl,divergence etc. have well defined precise notions for these fields.
We know electrostatic and ...
- 1,578