The four fundamental fundamental equations of electromagnetism.

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Lagrangian of classical electromagnetism without $A_{\mu}$ field [duplicate]

Is there a Lagrangian reproducing Maxwell's equations without the use of the scalar and vector potential? Obviously $\mathcal{L} = -\frac14F_{\mu \nu}F^{\mu \nu}$ doesn't work since in terms of $E$ ...
7
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4answers
821 views

Least-action classical electrodynamics without potentials

Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which ...
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1answer
29 views

How can a sinusoid be a steady current? [duplicate]

As far as I understand it, a steady/stationary/constant current is defined to have $dJ/dt=0$ (i.e., no explicit time dependence). So I would say that sinusoids cannot produce steady currents, yet ...
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1answer
39 views

Intuition differential ampere's law

Ampere's differential law states that - $$\nabla \times {\bf B} = \frac{4 \pi \, {\bf J}}{c}$$ I know to derive amperes integral form from special relativity, and to use stokes theorem in order to ...
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1answer
425 views

What are the boundary conditions for EM waves normally incident on the interface between two dielectric media?

An EM wave, amplitude $E_0$, frequency $\omega_0$, is incident upon a material with relative permittivity (dielectric function) $$\varepsilon \left( z \right) = \left\{ \begin{gathered}{\varepsilon ...
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0answers
57 views

Conformal invariance of Maxwell equation in presence of external current

It is known that pure electrodynamics in curved space-time is invariant under Weyl transformations $$ \tag{1} g_{\mu\nu} \to \Omega(x)g_{\mu\nu}, \quad F_{\mu \nu} \to \Omega^{-1}(x)F_{\mu\nu}. $$ ...
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1answer
44 views

About the closed line integral of electric field intensity

In electrostatics, we know that the closed line integral of electric field is zero : \begin{equation} \oint\limits_{C} \mathbf{E}\left(\mathbf{x}\right) \boldsymbol{\cdot} \mathrm{d}\mathbf{x}=\;\;...
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2answers
33 views

Why does current follow a conductor above a ground plane

Suppose there is a conductor above a ground plane. Current flows from a source through the conductor to a load on the other side. Depending on the frequency of the current the return path through the ...
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2answers
38 views

How do charges accumulate even though current flows through a capacitor?

I don't understand why do charge accumulate on each plate of capacitor.I learned about displacement current which flows through the gap of the capacitor and this makes the circuit continuous.But why ...
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0answers
26 views

wave equation in dielectric medium [on hold]

Maxwell's equations: $$\nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon}$$ $$\nabla \times \mathbf{E} = - \frac {\partial \mathbf{B}} {\partial t}$$ $$\nabla\cdot \mathbf{B} = 0$$ $$\nabla \times \...
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2answers
57 views

DC current in a wire

I'm sure that this question was addressed here before, but I failed to find any other instances, so with your permission I ask the question myself. I'm experiencing a very disturbing glitch, there is ...
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1answer
99 views

Does Maxwells equations imply $R=const*\rho$?

Suppose we have a resistor in a strange shape, filled with a medium of resistivity $\rho$, assuming only maxwells equations apply, is it true that R is proportional to rho, even for very low ...
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1answer
306 views

Surely force on shell can't be balanced by field momentum?

Imagine a particle with charge $q$ at rest at the origin. It is surrounded by a concentric spherical insulating shell, also at rest, with charge $Q$ and radius $R$. At time $t=0$ I apply a constant ...
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1answer
123 views

Is it true that $\frac{d}{dt}\int_S \mathbf{B} \cdot d \mathbf{a}$ goes to zero if the amperian loop delimiting $S$ contracts indefinitely?

I suppose to have an ordinary magnetic field: in the answer I'm not interested to involve Dirac delta: the integral goes to zero. I want to focus on another point: an infinitesimal physical quantity ...
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0answers
42 views

Defining electromagnetic stress tensor for non-linear media

In textbooks, the electromagnetic stress tensor (in vacuum also called Maxwell stress tensor) is usually derived for linear media, implying that $$ \vec D = \epsilon_0 \epsilon_r \vec E$$ My question ...
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1answer
158 views

Spherical magnet inside a solenoid

When passing a bar magnet through a long solenoid why is it that the induced emf when the magnet is in the middle of the solenoid is zero? And if a spherical magnet is put inside the solenoid, will ...
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0answers
32 views

Maxwells equations at a point

Why is it that whenever a problem is posed for maxwells equations -say the electric field at some area- that it is only requested for a fixed/given point? It would seem like you would want the field ...
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0answers
41 views

Deriving an equation in Maxwell's “a treatise on electricity and magnetism” [duplicate]

I am reading Maxwell's "a treatise on electricity and magnetism" and I need a derivation of formula 16 $\left(M=\iint\dfrac{\cos\varepsilon}{r}\mathrm ds~\mathrm ds'\right)$ (in the page below) using ...
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1answer
155 views

How can electric field representation be obtained from Enge representation using Maxwell's equations?

Suppose we have a long electric capacitor. Let $L$ be its length ($z$ coordinate), $W$ its width ($y$ coordinate), and $D$ its full height (full aperture; $x$ coordinate). Let $L\gg W\gg D$. The ...
6
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1answer
359 views

Confusion in Maxwell's derivation of Ampere's Force Law - Part II [closed]

I am reading Maxwell's "a treatise on electricity and magnetism, Volume 2, page 156" about "Ampere's Force Law". I have some confusion in the following pages: My question is of two parts: 1. ...
6
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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 ...
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1answer
40 views

Rewriting Maxwell's equation in tensor form [closed]

Suppose $F_{ij}=\epsilon_{ijk}B_k $, how to prove the following: $\partial_iB_i=0$ becomes $\partial_iF_{jk}+\partial_jF_{ki}+\partial_kF_{ij}=0$ $B_iB_i$ becomes $F_{ij}F_{ij}/2$ I can see that ...
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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 ...
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4answers
671 views

How would you define electrostatics and magnetostatics starting from Maxwell's equations?

I'm reading Griffith's text, and he starts by defining Electrostatics as requiring the source charges don't move. I've seen a few slightly different definitions of electrostatics and magnetostatics. ...
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22 views

What is the physical meaning of a magnetic conduction current?

In electrodynamics, it is possible to have an electric conduction current, whereby $J=\sigma_e E$, with $J$ being the current, $\sigma_e$ the electrical conductivity and $E$ the electric field (this ...
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2answers
567 views

Why do we use gauges in Maxwell equation?

While solving the Maxwell's equation we often use the Lorenz or Coulomb gauge, but why is that? Are the equations unsolvable if the gauge is not fixed? Or is it just for the simplicity?
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3answers
924 views

What are the differences between the differential and integral forms of (e.g. Maxwell's) equations?

I would like to understand what has to be differential and integral form of the same function, for example the famous equations of James Clerk Maxwell: How to know where to apply each way? Excuse ...
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1answer
60 views

Tensor notation of Maxwell's equations

Tensor notation of Maxwell's equation read So when we explicitly try to find the Maxwell's equation from the above tensor equation we only get gauss law and curl of B. The div.B=0 and curl of E are ...
5
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1answer
935 views

Maxwell equations and symmetry

Do the full inhomogeneous Maxwell equations obey parity (P) and time reversal (T) symmetry separately or only the full CPT symmetry? I believe the homogeneous Maxwell equations obey parity and time ...
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1answer
44 views

Interpreting $\hat{e}_z$ in Maxwell's equations

I'm trying to interpret a form of Maxwell's equations, but I can't seem to figure out where the term $\hat{e}_z$ comes from in the following equation: $ \frac{\partial{\vec{E}_t}}{\partial{z}}+i\frac{\...
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32 views

Propagation Of Wave in Rectangular Waveguide

From what I understand, electric and magnetic fields are perpendicular to one another and the direction of wave propagation.The text book states that the direction of wave propagation in the ...
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2answers
196 views

How can Maxwell theory be viewed in terms of two-layer structure?

I'm trying to learn more about Maxwell equations and stumbled upon an essay by professor Freeman J. Dyson from Princeton. He explained Maxwell theory in a very interesting way. The modem view of ...
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3answers
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Deriving the speed of the propagation of a change in the Electromagnetic Field from Maxwell's Equations

I've been told that, from Maxwell's equations, one can find that the propagation of change in the Electromagnetic Field travels at a speed $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ (the values of which can ...
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1answer
95 views

How to solve “EM wave equation” for the field of uniformly moving charge?

Is it possible to show that the field of a uniformly moving charge, which is according to Biot-Savart law is given by... $${\bf E}({\bf r},t)=kq\left(\frac{1-v^2/c^2}{(1-v^2 \sin^2 \theta/c^2)^{3/2}}\...
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Derivation of Poynting theorem in matter

In most textbooks I have read they derive the Poynting theorem using the Maxwell's Equation in vacuum and the fact that the force density $f=\pmb{E} \cdot \pmb{J}$. Then they just generalize it ...
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How is the curl of the electric field possible?

Taking the curl of the electric field must be possible, because Faraday's law involves it: $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$ But I've just looked on Wikipedia, where it ...
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3answers
683 views

How to derive the expression for the electric field in terms of the potential?

How can I derive that $$\vec{E}=-\vec{\nabla}\phi-\frac{\partial \vec{A}}{\partial t}$$ where $\phi$ is the scalar potential and $\vec{A}$ the vector potential?
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Why does a 2-sided propagating EM wave become 1-sided if B is made proportional to E?

If you simulate the propagation of an electromagnetic wave in 1D free space (no charges or currents) with initial conditions of $E\neq0$ and $B=0$, and you look at a movie of $E$ vs time, then after ...
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30 views

Problem with understanding boundary conditions in electromagnetism

In some books on electrodynamics they stress that electric current won't radiate if it is placed on a perfect electrical conductor (PEC), citing image theory: exactly opposite current will appear and ...
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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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3answers
177 views

Maxwell's equations - underdetermined - uniqueness

Maxwell's equations can be seen as two dynamical equations (the two curl equations), and two constraint equations (the two divergence equations). So we have 6 unknowns ($E_x,E_y,E_z,B_x,B_y,B_z$). ...
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7answers
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Do Maxwell's Equations overdetermine the electric and magnetic fields?

Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\vec{E}=(E_x,E_y,E_z)$ and $\vec{B}=(B_x,B_y,B_z)$...
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2answers
234 views

Boundary conditions for Maxwell's equations at the interface between two media

Consider the following simple Maxwell's equations: $$ \nabla\cdot\mathrm{D}=\rho $$ $$ \nabla\times\mathrm{E}+i\omega\mathrm{B}=0 $$ $$ \nabla\cdot\mathrm{B}=0 $$ $$ \nabla\times\mathrm{H}=\mathrm{J}+...
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1answer
75 views

What causes electromagnetic waves to propagate in free space?

In free space, $\rho=0$ and $J=0$, so there are no electromagnetic sources/sinks. Maxwell's equations thus reduce to: $\nabla\cdot E = 0$ $\nabla\cdot B = 0$ $\nabla\times E = -\frac{\partial B}...
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1answer
55 views

Derivatives involving four vectors [closed]

The Schrödinger lagrangian for complex fields is $$L=\frac{1}{2m}(D_i \psi)^* Di \psi - \frac{i}{2} \left[\psi ^* D_0 \psi - (D_o \psi)^* \right] - \frac{1}{4}F^{\mu \nu}F_{\mu \nu}$$ Where $D_\...
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How do the mode expansion of the $A_\mu$ field satisfy Maxwell's equations?

I want to show that the mode expansion $$A^\mu(x)=\int\frac{d^3\vec{p}}{(2\pi)^32E_\vec{p}}\sum_r\left[\epsilon^\mu_r(\vec{p})a_r(\vec{p})e^{-ip\cdot{x}}+\epsilon^{\mu*}_r(\vec{p})a^\dagger_r(\vec{p})...
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21 views

Which formulas would tell me the gradient of an electromagnetic field at an arbitrary distance from a pole? [duplicate]

I'm a newbie to physics and was wondering where I can read about electromagnetic gradients. From what I understand (and my intuition) electromagnetic fields create force gradients around its poles. ...
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4answers
2k views

What could magnetic monopoles do that electrically charged particles can't?

I understand the significance to physics, but what can a magnetic monopole be used for assuming we could free them from spin ice and put them to work? What would be a magnetic version of electricity? ...
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1answer
54 views

Questions about Biot-Savart law and Ampere's law

A textbook I'm studying with described finding vector magnetic potential $\vec{\text{A}}$ from Biot-Savart law as below. $\vec{\text{H}_2}=\int_{\text{vol}}\frac{\vec{\text{J}_1}\times\hat{\text{a}...
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19 views

wave propagation modelling

what is the best modelling technique for modelling mm-wave propagation in electromagnetic environment. Right now,am working on how to use use Transmission-line matrix (TLM) and ray-tracing techniques