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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Why dielectric polarization is not considered at boundary condition?

Consider the problem of reflection, transmission for incident light at the boundary of two dielectrics. From Ampere' law, $\vec\nabla\times \vec H = \vec J+\partial\vec D/\partial t $. Every text ...
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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 \...
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Could magnetic fields really be completely substituted by relativity and electric fields?

In many textbooks (especially those for undergraduate level), magnetic fields are described merely as a relativistic side product of electric fields when considering frames in motion relative to ...
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Questions about plane electromagnetic wave

A wavefront is defined to be the locus of all points on which the field oscillations are in phase, in the plane electromagnetic wavefront, it is said that all points on a plane perpendicular to the ...
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What is knotted in EM and GR?

I found this paper with beautiful illustrations: Helicity, Topology and Kelvin Waves in reconnecting quantum knots, and this one which seems to describe something closely analogous: New knotted ...
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Why Proca Term forbidden in Schwinger Model?

In my QFT Lecture we considered the Schwinger model with a Proca term. Solving the eom for the Stueckelberg field and plugging it back into the original Lagrangian, we receive an effective Lagrangian ...
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Ampere's Law, Interface conditions for magnetic field

I'm failing to understand the derivation of the interface conditions for the tangential components of the magnetic field given her (based on d.j,griffiths) Ampere's law in integral form is given as $$...
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The curious case of parallel $E$ and $B$ fields and inertial frames

In a comment to this Physics SE question, @MichaelSeifert stated, For the more general case, IIRC there's always a frame in which $\vec{E}$ and $\vec{B}$ are parallel when $\vec{E}\cdot \vec{B}\...
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Covariant Maxwell equations invariant under parity transformation

I tried to proof that the Maxwell equations are invariant under parity transformations. Therefore I used the covariant formulation of the Maxwell equations \begin{align} \partial_{\nu}F^{\nu\mu} &...
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Maxwell's equations in differential form in 2-space+1-time dimensions

How does one write maxwell's equation in 2+1 dimensions? It becomes particularly interesting as the components of 2 forms and 1 form are 3. Are there any sources for this?
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What is the formula for electromagnetic force on magnetic charge in condensed matter?

It is known that if we take into account the magnetic charges, Maxwell's equations acquire a symmetrical form (Jackson, 3rd edition, eq. 6.150): \begin{align} \begin{aligned} -\nabla \times \...
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Does the induced current in one wire affect the driven wire?

Let's say that we have two parallel, vertical wires with radii much smaller than lengths. The wire on the right is driven with an AC current. When the current is increasing in the upwards direction a ...
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Numerically Stable Light Absorption Density Calculation

I am reading through the documentation provided by Lumerical concerning light absorption per unit volume : https://kb.lumerical.com/layout_analysis_pabs_simple.html Lumerical says that: It can be ...
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Is there Electric Flux through a closed loop around a current wire?

In one of his lectures, Professor Walter Lewin is ammending Ampère's Law to include displacement current (i.e. It not only depends on the current that penetrates the loop, but also on the changing ...
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Difference between $\bf J$ and time derivative of $\bf E$ in Maxwells equations? [closed]

Maybe I am being confused. It was some years ago I did this. An electric current changes charge distribution which creates rotation in $\bf B$. So in Ampères / Biot-Savarts law what is the difference ...
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Maxwell-Chern-Simons equation: Translating from differential form to component form

I am trying to solve the scalar-coupled Maxwell-CS equations (which is one of the equation of motions in $N=2$ supergravity coupled to 3 vector multiplets), which is written in this form in the ...
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Field intensity for electric field and vector potential

In general, the intensity of an electric field is given by $$ I = \frac{c\epsilon_0}{2}E_0^2 $$ where $E_0$ is the peak amplitude of the electric field. Let's say we have an electric field $$ E(t) = ...
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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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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 ...