The four fundamental fundamental equations of electromagnetism.

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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}}...
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2answers
71 views

What does it mean to be unique in terms of vector potentials?

I was in an electromagnetism lecture, where we were looking at the magnetostatic Maxwell’s equations: $$\begin{align} \nabla\cdot\mathbf{B} &= 0 \\ \nabla\times\mathbf{B} &= \mu_0\mathbf{J} \...
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0answers
38 views

What does this coordinate transformation in the Wave equation mean?

My tutor derived the following of which I do not understand the transformations (2.1) and (2.2): $$\Delta\vec{E} - \frac{1}{c^{2}} \frac{\partial^{2}\vec{E}}{\partial t^{2}} = \frac{4\pi}{c^{2}} \...
3
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0answers
81 views

Why must this boundary condition be met? (Electromagnetic wave at interface between two mediums)

My textbook says that The laws of Electromagnetic Theory (Section 3.1) lead to certain requirements that must be met by the fields, and they are referred to as the boundary conditions. ...
2
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0answers
82 views

Do divergence and curl of Lorentz force have some physical meaning?

Time ago I started thinking about this: if we take the well known Lorentz Force expression, namely $$\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)$$ and we operate $\nabla\cdot \...
5
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0answers
80 views

Maxwell Equations don't give unique Electric Field?

Consider the class of electric fields given by $$\mathbf{E}=\begin{cases} \ln (Cr)\hat{z} & 0\leq r < R\\ 0 & r> R \end{cases}$$ where $C$ is a constant and $r$ is the polar-distance ...
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0answers
22 views

Time dependance of oscillating sheet of charge

I am working on a practice problem involving maxwells equations. We have an infinite sheet of charge density $\sigma$ in the x-y plane and it is oscillating as x= $\Re [x_0e^{-i\omega t}]$. I want to ...
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1answer
52 views

Electric Displacement Vector

How do I interpret what electric displacement vector is? I know that it exists and I know it's an equation but I'm not able to really understand or interpret what it is. $$\oint_A \mathbf{...
0
votes
1answer
82 views

About curious form of Maxwell's Equations for a monochromatic field [closed]

In a review paper of Whispering-gallery waves from A.N. Oraevsky, he writes the source-free monochromatic Maxwell's Equations as $\nabla\times E = ikH$ $\nabla\times H = -ikE$ and he defines $k = (\...
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42 views

Displacement current in a capacitor

I recently learned about the concept of displacement current in a capacitor. I've understood the basics - mainly understood the reason why it was introduced. However, is it purely a mathematical ...
4
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2answers
142 views

How can one meaningfully say that one field generates the other in an EM-wave?

This is a follow up question to: Do the electric and magnetic components of an electromagnetic wave really generate each other? Clearly there are nuances of how one states the "mutual induction" ...
6
votes
1answer
414 views

Biot-Savart law from Ampère's with multivariate calculus

Let us assume the validity of Ampère's circuital law $$\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}$$where $\mathbf{B}$ is the magnetic field, $\gamma$ a closed path linking the ...
0
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1answer
43 views

Relation between displacement current, dielectric and time varying Electric field

I know that displacement current is produced in dielectric material due to dipole moment. I also know that displacement current is produced by time varying electric field (according to maxwell ...
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2answers
64 views

How does one show Maxwell's equations in vector calculus form describe the same motion in all reference frames?

The covariant form of Maxwell's equations is Lorentz invariant. $$\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta}$$ $$\partial_{\alpha}F_{\beta\gamma} + \partial_{\beta}F_{\gamma \alpha} + \...
2
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1answer
155 views

Ampère's law from Biot-Savart law for linear currents with multivariate calculus

My book, W.E. Gettys's Physics, starts from the Biot-Savart law $d\mathbf{B}=\frac{\mu_0}{4\pi}\frac{Id\boldsymbol{\ell}\times\hat{\mathbf{r}}}{r^2}$, i.e.$$\mathbf{B}(\mathbf{x})=\frac{\mu_0}{4\pi}\...
4
votes
1answer
118 views

Why is the magnetic field of a spherically symmetric current zero?

We now ask about the magnetic field produced by the currents in this situation. Suppose we draw some loop $\Gamma$ on a sphere of radius $r,$ as shown in Fig. 18–1. There is some current through this ...
18
votes
1answer
543 views

Do light waves precisely follow null geodesic paths in General Relativity?

In special relativity one may show that a plane wave solution of Maxwell's equations (in a vacuum), of the form $A^a=C^a\mathrm{e}^{\mathrm{i}\psi}$ has the following properties: The normal $k:=\...
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0answers
54 views

When can I use Helmholtz equation for electraomagnetic wave

The complete Maxwell wave equation for electromagnetic field using the double curl operator "∇×∇×". Only when the transverse condition is hold, this operator can equal to the Laplace operator and form ...
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0answers
26 views

Purpose of magnetic charge and current density

I am aware that introducing a magnetic charge density and a magnetic current density makes Maxwell's equations much more symmetric. But in what situations/problems is this beneficial? Could someone ...
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0answers
46 views

Deriving Maxwell's equation from the Lagrangian of an electromagnetic field with a charge density $\rho$

I want to derive Maxwell's equation from the Lagrangian of an electromagnetic field with a charge density of $\rho$ The Lagrangian is given by $$L={1\over2}(\epsilon_0E^2-{1\over\mu}B^2)-\rho\phi+\...
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0answers
39 views

Can we prove that EM waves are transverse from Gauss's law of magnetism? [duplicate]

Can we use $ \mathbf B \cdot d\mathbf s=0$ to prove transverse nature of EM waves?
4
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4answers
107 views

About the uniqueness of the displacement current

In the Maxwell-Ampère equation, i.e.: \begin{equation} \nabla\times\vec{B} = \mu_0 \vec{J} + \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} \end{equation} the $\vec{J}_d$ term: $$ \vec{J}_d := \...
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2answers
141 views

Problem related to application of Maxwell's equation for point charge moving uniformly

Maxwell's 4th equation which describes magnetic field, has two terms: $$ \oint \mathbf{B}\cdot d\mathbf{l}=\mu I+\mu \varepsilon \frac{\mathrm{d}\Phi}{\mathrm{d}t}$$ Now, I wanted to derive the ...
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votes
1answer
91 views

How to draw + and - and $\nabla \times E$ on a circular wire?

Faraday's Law: $$\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$$ Circulation of electric field: For time-varying magnetic field and a closed wire, How can we add + and - pole ...
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votes
4answers
216 views

Why should gluons move at a speed determined by $\mu_0$ and $\varepsilon_0$?

I understand that the speed of light can be derived from Maxwell's equations, giving $c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}$ I furthermore understand how the principle of invariance of laws w.r.t. ...
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1answer
65 views

What are some experiments verifying Maxwell's 4th equation?

Please give me the link to some paper, website or book that you know about which discusses in detail some experiments which quantitatively verify Maxwell's 4th equation, which is $$\quad \nabla\times{...
3
votes
1answer
161 views

How is Biot-Savart law verification of Maxwell's 4th equation for steady current?

Please provide some theoretical procedure which equates Biot-Savart law with the Maxwell's 4th equation for steady current, which is Ampere's law $$\quad \nabla\times{\bf B} = \mu_0{\bf J}.$$
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0answers
81 views

Magnetic monopole using differential forms

I'm trying to understand the different variations of the Maxwell's equations using differential forms. The Maxwell's equations are $$dF=0\\ *d*F=J$$ where $F$ is the electromagnetic tensor ($F=dA$) ...
0
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1answer
82 views

Maxwell equation correct statements [closed]

Do these four statements correctly define the 4 Maxwell's equations or do I need to add or change something in it ? 1)Gauss's law: The net electric flux passing through any closed surface is $\large \...
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0answers
102 views

Can a Set of “Maxwell's Equations” for Newtonian Gravitation be Derived from Newton's Force + Special Relativity?

When I learned about electromagnetism in my first year of undergraduate school, Maxwell's equations were derived roughly in the following way (see also here or in [1]): Gauss's law for a static ...
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1answer
112 views

Proof that Maxwell equations are Lorentz invariant

In Peskin and Schroeder page 37, it is written that Using vector and tensor fields, we can write a variety of Lorentz-invariant equations. Criteria for Lorentz invariance: In general, any equation ...
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1answer
33 views

magnetic force from v=0 and dB changing

magnetic force is defined as: $ \mathbf{F}=q \mathbf{v} \times \mathbf{B}$ so if v=0 there is no force. However Faraday's law states that if $d\mathbf{B}\neq0 $ then we have an induced emf even ...
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0answers
31 views

derivation of the electic displacement D

I've been revising Maxwell equations recently and tried to prove that the electric displacement $ \mathrm{\nabla \cdot D = 0}$ in the electrostatic approximation ($\mathrm{E = - \nabla} f$). My steps ...
0
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1answer
92 views

Maxwell's Equation

I was reading a book and I found the following equation. I'm not good at physics and I couldn't understand where does it come from, can anyone explain me why is it valid? $$ \mathbf E = -\nabla\phi - \...
0
votes
1answer
108 views

Maxwell equations in 2+1 D

I have a problem with the Maxwell equations in (2+1) dimensions using differential form. Following J. Baez "Gauge Fields, Knots and Gravity" page 93 (or any other book), the equations are \begin{...
4
votes
1answer
58 views

Why do these calculations of EM fields for a magnet and wire loop seem inconsistent?

Suppose you have a conducting circular wire loop and a magnet moving towards each other. They move along the $z$ direction with nonrelativistic constant speed $v$. Let the $B$ field of the magnet in ...
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0answers
19 views

How Maxwell's equations are derived from fundamental equations for electrostatic and magnetostatic models [duplicate]

How Maxwell's equations are derived from fundamental equations for electrostatic and magnetostatic models.
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0answers
41 views

RF cavity Field Maps using Poisson Superfish

This may be a slightly obscure question but I'm trying to use Poisson Superfish to generate field maps for single cell of an ILC SRF Cavity. There are examples in the documentation for radio ...
1
vote
1answer
122 views

Motion of Thompson's jumping ring

Thompson's jumping ring experiment is set up as follows: There is a force acting on the ring $F(x)$ where $x$ is the vertical displacement. The force is due to the $90^\circ$ out of phase flux ...
1
vote
1answer
36 views

What are some of the empirical proofs of electromagnetic polarization? [closed]

I am aware of how polarization follows from Maxwell's equations, and how it is possible in transverse waves in general. I also know that Huygens, in his great Treatise on Light, first discovered and ...
5
votes
1answer
178 views

Why do magnetic field lines describe a force?

My professor stated the four Maxwell equations, as well as the "Lorentz force" equation $$ \mathbf{F} = q\left(\mathbf{E}+\frac{1}{c}\mathbf{v} \wedge\mathbf{B}\right) \tag{1} $$ He said that this ...
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1answer
49 views

Demonstration for the existence of longitudinal electrostatic oscillations

How could I demonstrate that in a linear, homogeneous and isotropic medium without losses but electrically charged, Maxwell's equation admit as solutions longitudinal electromagnetic waves, beyond ...
1
vote
1answer
57 views

Determining the phase delay between H and E fields

I want to determine the phase delay between H and E fields in a medium with losses and not electrically charged. In this medium we also have $\sigma_c\approx \varepsilon \omega$. The enunciation of ...
0
votes
1answer
132 views

how do photons move with respect to EM (I'd like to picture wave magnitudes frame by frame) [duplicate]

(I'm aware treating photon as particle and talking about its position is not exactly, conceptually right but I think it makes sense, at least in the point of view of a beginner. Please just assume ...
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0answers
62 views

Distribution of current, charge and EM radiation from a centrally AC fed ball-shaped antenna

As far as I can understand, it is generally accepted that every classical electromagnetic phenomena can be explained by five equations: Maxwell's four equations and the Lorentz force law. However you ...
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26 views

Effective Medium

Please consider the following problem : A plane wave of wavenumber k is incident on an infinite slab which is inhomogeneous in the z direction. Also assume harmonic time dependence and that the ...
0
votes
3answers
177 views

How to propagate $E_x (x,t) = \exp(-t^2/\tau^2-i\omega_0 t) \exp(-x^2/w_0^2)$ in finite difference time domain (FDTD) analysis

Finite difference time domain (FDTD) allows to solve differential equations for time evolution. For example, we can analyze ultra-short pulses in free space by solving the Maxwell's equations. The ...
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votes
1answer
66 views

How far will a 1Hz EM Wave propagate if it's source oscillator is running for exactly one second? [closed]

If you have an Oscillator set on 1Hz and you let it run for exactly one second connected to an Antenna , how far will the generated EM Wave travel ?
7
votes
1answer
559 views

Do photons decay as they travel in free space

From maxwell's equations, it occurred to me that photons are stable. Decrease in electric field creates magnetic field and vice versa and somehow there is a harmony that allows photon to exist as long ...
3
votes
1answer
226 views

Why does a function $\psi(v)$ appear when a Lorentz transformation is applied on Maxwell's equations?

In his 1905 paper, Einstein applies Lorentz transformation on Maxwell's equations, and relates the electric force $(X,Y,Z)$ and magnetic force $(L,M,N)$ in an inertial frame $K$ with spectime ...