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Questions tagged [maxwell-equations]

A set of four equations that define electrodynamics. They comprise the Gauss laws for the electric and magnetic fields, the Faraday law, and the Ampère law. Together, these equations uniquely determine the electric and magnetic fields of a physical system. Do not use this tag for the thermodynamical equations known as Maxwell's relations.

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Faraday's Induction Law Notation

I am confused as to the notation used in a course I'm taking on physical optics. I have presented 2 variants of Faraday's Law, combined with the full set of Maxwell's equations. The first ...
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Physical Interpretation of $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} $

The differential's form of Gauss' Law is $$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}. $$ This suggests that at every point in space, the the electric field $\vec{E}$ is determined by the charge ...
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Electric field produced by a moving charged particle above a planar dielectric interface

The electrostatic field of a single charged particle above a planar dielectric interface is a standard example given in many books (see example 4.4 in Griffiths or https://en.wikipedia.org/wiki/...
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Current induced by permanent magnet moving along toroid coil

(It's not really shown in the pictures, but please assume that the coils are included in a closed circuit in both configurations. Maybe they could serve as a tension source in their respective circuit ...
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How correct is Randell Mill's “The grand unifed theory of classical quantum mechanics”? [on hold]

In https://doi.org/10.1016/S0360-3199(01)00144-6 , the author has claimed to have unified Maxwell’s equations, Planck’s equation, the de Broglie equation, Newton’s laws, and special, and general ...
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Ampère's circuital law in case of uniform current density in infinite space

Let's have an ideal conductor with current density $\mathbf{J}$. The ideal conductor takes up the entire space, effectively resulting in the entire space $\mathbb{R}^3$ being permeated with $\mathbf J$...
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Electrodynamics of inhomogeneous media

I'm interested in electromagnetic scattering and for that purpose I would like to write out Maxwell's equations for space-dependent permittivity (and permeability). Actually I only need them for ...
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Equivalence between Maxwell's equations and vector Helmholtz equations

When are equivalent the Maxwell's harmonic equations: $$ \nabla\times\left(\nabla\times\mathbf{E}\right)=\mu\epsilon\omega^2\mathbf{E} $$ and the vector Helmholtz equations: $$ \nabla^2\mathbf{E}=\mu\...
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Explanation of Lenz's Law phenomena

If we drop a magnet through a copper pipe (without it touching any of the sides), it would fall slower than it would if there were no pipe. Having the pipe otherwise accelerate the magnet would be in ...
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How was the value of vacuum permittivity originally found?

The vacuum permittivity appears originally in Maxwell's equations, used to describe electric fields. The permeability of vacuum was defined using Ampere's force law (itself derived from Biot-Savart ...
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How does Superposition principle follow from Maxwell's equation's linearity?

It is said that whole of electromagnetism can be completely described by the Maxwell's equations. The thing that intrigues me is that how does superposition principle follow? First, I take an ...
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are there changing magnetic and electric fields that are not EM radiation?

Let us consider these two Maxwell equations: $$\frac{\partial \vec{B}}{\partial t}=-\vec{\nabla}\times \vec{E}$$ and $$\frac{\partial \vec{E}}{\partial t}=\frac{1}{\epsilon_0}\left(-\vec{J}+\frac{1}{...
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Are Maxwell's equations “physical”?

The canonical Maxwell's equations are derivable from the Lagrangian $${\cal L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$ by solving the Euler-Lagrange equations. However: The Lagrangian above is ...
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Flattening Electrodynamics in a curved space

It is possible, apparently, to describe gravitational lensing as if gravitational potential induces an effective refractive index change in the vacuum, and spacetime is flat. As pointed out by @...
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Electrodynamics confusion - Hertzian dipole

I am studying a course in Electrodynamics and we are just covering retarded potentials and the Hertzian dipole. In my lecture notes, we have calculated the magnetic vector potential $A$ in the ...
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Different predictions from differential vs integral form of the Maxwell–Faraday equation?

Assume a toroidal solenoid with a variable magnetic field inside (and zero outside) and a circular wire around one of the sides. Because there is no magnetic field outside the solenoid, we have $$\...
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Gravitational potential, effective refractive index, and vacuum charge density

In an earlier question, I asked about how to explain gravitational lensing to a layman in terms of propagating wave fronts, in a way analogous to the way an optical lens can be explained: the ...
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Is the Dirac Lagrangian locally gauge invariant without gauge field $A$?

When it comes to the check of the invariance of the Lagrangian of the Dirac equation under local $U(1)$-transformations I have made the following observation: $$L = \bar{\psi} (i\gamma^{\mu}\...
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Induced EMF in single stationary wire

Suppose we have a conducting stationary wire in a uniform magnetic field: $$\mathbf B(t) = kt \mathbf u_z$$ with $k>0$. Assume the wire is a segment that lies on the $xy$ plane and its length is ...
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Displacement current derivation

A common explanation for the reason why displacement current is needed is in the following diagram (Giancoli): I can appreciate the reason why we need displacement current, however I really don't get ...
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Comparison of covariant form of Maxwell equations with Einstein's GR

We know, the the vector form of Maxwell equations \begin{align} \vec\nabla\cdot\vec{E} &= 4\pi\rho \label{Diff I}\\ \vec\nabla\times\vec{B} &= \dfrac{4\pi}{c} \vec{j}+\dfrac{1}{c}\dfrac{\...
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Why do we assume simply connected domains and continuously differentiable fields in electromagnetism theory?

In many textbooks, including Griffiths', they erroneously claim that a field is irrotational if and only if it is conservative (there exists a scalar potential). This is true only if the domain of ...
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If Ampere's law implies the Biot-Savart law, which implies Gauss's law for magnetism, does that mean Maxwell's equations are redundant?

Studying electromagnetism, I came across the following fact: Maxwell's third equation (divergence of magnetic field is zero) can be derived from the Biot-Savart Law. The Biot-Savart Law can be ...
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The relationship between material properties and EM wave frequency

Assuming an EM wave traveling inside an electrically neutral dielectric material. The following electric field describes the EM. $\vec{E}(t,x)=20\cos(\omega t-50x)\vec{u_y}$. Using the 3rd Maxwell ...
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Derivation of Covariant Maxwell's Equations

I am trying to derive the covariant formulation of Maxwell's equations. I understand that all four of Maxwell's equations can be written compactly as $$\partial_{\mu}F^{\mu\nu} - j^{\mu} = 0 \;, \...
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Problem with faraday's law in a closed circuit with a battery: is there a changing B flux?

We know that in a closed circuit connected to a battery, $\oint E.dl\ne0$, due to the non conservative nature fo the EMF generated by the battery. But, according to Faraday's law, then $\int_{\Sigma}\...
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Ampères law: Getting $\nabla \times \vec H = \vec J_{free} + \frac{\partial \vec D}{\partial t}$ by taking the cross product

I've seen two different versions of Ampère's law and I'm having trouble connecting them: $$\nabla \times \vec B = \mu_0 \vec J + \mu_0 \epsilon_0 \frac{\partial \vec E}{\partial t}$$ $$\nabla \...
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Maxwell theory : How to justify the potential gauge fields if there are magnetic monopoles?

Without magnetic monopoles, the Maxwell equations are these (I'm dropping vector notations and all constants, for simplicity): \begin{align} \nabla \cdot E &= \rho_{elec}, \tag{1} \\[12pt] \nabla \...
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Relation of dispersion for a plasma

Assuming the electric field : $\vec{E} = E_{0}\,e^{i(kz-\omega t)}\vec{e_{z}}$ and the complex relation by doing $\vec{rot}\,(\vec{rot}\times\vec{E})$ with $\vec{rot}\times \vec{E}= i \vec{k}\...
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Magnetic field at points on the circuit

I know magnetic field lines due to a circuit always form closed loops. Therefore $\nabla \cdot \vec{B}=0$ everywhere (even at points on the circuit). However due to singularity, magnetic fields are ...
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KVL for non- conservative E-field

Can we use KVL in a circuit having non-conservative field. I mean if its true then it denies the Maxwell equations which says that closed loop integral of E.dl is not zero in non-conservative fields.
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Do Maxwell's equations predict the speed of light exactly?

I know that $\frac{1}{\sqrt{\mu_0\varepsilon_0}}$ is equal to the speed of light but is this prediction accurate? I mean is it 100 percent accurate?
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Maxwell-Faraday Equation and change in magnetic flux

Is the change in flux being equal to negative emf an experimental law? The Wikipedia derivation of emf as a negative change in magnetic flux in time: https://en.wikipedia.org/wiki/Faraday%...
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Experimental evidences satisfying charged particle emits radiation in gravitational field

Is there any experimental evidence exist that a charged particle at rest in gravitational field emit radiation and charged particle in free fall don't emit radiation ( According to equivalence ...
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How much of Maxwell's equations is recoverable from the zero divergence of the stress-energy tensor?

As a motivating example, consider the static electromagnetic field defined by $\textbf{E}=(\text{const})x\hat{\textbf{y}}$, $\textbf{B}=0$. The stress-energy tensor for this field is $T=\operatorname{...
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Why does Maxwell's equations $\partial_{\mu} F^{\mu \nu} = 0$ have 3 independent components (DOF) in $D = 4$?

And how can we generalize this to the statement that it has $D-1$ independent components in dimension $D$? I know that $F_{\mu \nu}$ has six independent components (because of antisymmetry), how do ...
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If $A^\mu$ is not determined uniquely by Maxwell's equations, what happens if we solve for it numerically?

Given a solution $A^{\mu}(x)$ to Maxwell's equations \begin{equation} \Box A^{\mu}(x)-\partial^{\mu}\partial_{\nu}A^{\nu}=0\tag{1} \end{equation} which also satisfies some specified initial conditions ...
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How to derive the formula of angular momentum of light in Maxwell equation?

According to wikipedia,the angular momentum of light is expressed by $$\epsilon_0\int \left(E\times \vec{A} + \sum_{i=x,y,z}\vec E_i(\vec r\times \vec \nabla)A_i \right) d\vec r$$ How to derive this?...
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Maxwell Lagrangian where $F$ and its derivatives are the variables (i.e., without replacing $F={\rm d}A$) [duplicate]

The way the electromagnetic Lagrangian is usually constructed is by noticing that the EM fields are always constrained to satisfy ${\rm d}F=0$ (half of Maxwell's equations). We can immediately solve ...
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Why is FDTD derived directly from Maxwell's equations instead of the wave equation

I've been wondering why the Finite Difference Time Domain Method is derived directly from Maxwell's equations and not directly from the electromagnetic wave equation (that in theory is also derived ...
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What is the potential inside a hollow conducting sphere with multipoles uniformly surrounding it?

If considering a hollow conducting sphere with a surrounding uniform charge distribution, for example, it will have a constant and uniform potential throughout the inside of the hollow sphere because $...
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How is Maxwell's second equation true here?

$\vec{B}=\nabla \times \vec{A}\tag1$ This is true because at every point $\nabla\cdot\vec{B}=0 \tag2$ In free space points, $\displaystyle \vec{B}=\dfrac{\mu_0}{4 \pi}\int_C \dfrac{I\ dl \times\hat{...
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Can Maxwells equations be written as one equation?

Maxwells equations are usually written as a system of four equations. I've read somewhere that when Maxwell first wrote out his equations they were as a system of twenty equations though I think ...
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Maxwell Equations in Friedman-Robertson-Walker metric

The Maxwell equations are relativistic. But what happens to them in an expanding space time? I assume that only the charge density $\rho$ is affected, i.e. only Gauss's law gets modified. Am I right ...
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How should MTW's derivation of the Maxwell-Faraday formula be interpreted?

In the following derivation, I am not sure exactly how the components of the final vector equation are established. I suspect this is a situation analogous to the vector addition of infinitesimal ...
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How is the third case obeying integral form of Maxwell's second equation?

Let $m$ denote pole strength. In the diagrams: (1) Sky blue: Closed Gaussian surface (2) Red: North pole of magnet (3) Green: South pole of magnet (4) Yellow: Part of magnet cutting Gaussian surface ...
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Are Maxwell's equations valid in a rotating frame?

Maxwell's equations are covariant under Lorentz transformations. Are they covariant under going to a rotating frame and if not how do they look?
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How does a time varying magnetic field induce an electric field?

I've been trying really hard to understand this phenomena. Just exactly what happens in the region of space swarmed by this changing magnetic field that creates non electrostatic electric fields, ...
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Feynman Lectures Vol II-18 A travelling field: I'm not getting Feynman's result

I may just need to sleep on this, but I am not able to make sense of section 4 of The Feynman Lectures Vol II 18 The Maxwell Equations. After explaining the origin and meaning of the displacement ...
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Is Quantization implied in Maxwell's theory? [closed]

So as per Maxwell an electric charge oscillating at a certain frequency emits and absorbs radiations only of that frequency. So is quantization somehow implied here.? And I think Max Plank used the ...