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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How do I show that the two definitions of the curl of a vector field equal each other? [migrated]

The curl of a 3D vector field is a 3D vector itself and has two definitions - one in integral form and one in differential form. Definition 1: $$ \operatorname{curl}\vec{F}(x,y,z) \, \cdot \, \hat{n} ...
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Retarded Coulomb's law and EM waves; Feynman texts

In this chapter, Feynman writes down the retarded Coulomb's law, $$\begin{equation} \label{Eq:II:21:1} \mathbf E=\frac{q}{4\pi\epsilon_0}\biggl[ \frac{\mathbf e_{r'}}{r'^2}+\frac{r'}{c}\,\frac{d}{dt}\...
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Stumped on understanding a Feynman lecture about force from wire on magnet

I must (sheepishly) admit that I'm stumped on a beginning page of The Feynman Lectures Volume 2. I have included a picture from the page. [Let me know if I'm breaking copyright, or if I can include ...
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Deriving the speed of light from Maxwell's equations?

Relationship between speed of light and EM force? Can it be said that Maxwell used measurements of the "strength of electric force and strength of magnetic force", to derive the value for the speed ...
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Maxwell's equations in relativistic notation

We define $$x^\alpha=(ct,x,y,z)$$ $$\partial_\alpha=\frac\partial {\partial x^\alpha}=\biggl(\frac 1 {c^2} \frac\partial {\partial t} ,\nabla \biggr)$$ $$F_{\alpha \beta}=\begin{pmatrix} 0 & -\...
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Trouble understanding fractional flux linkage

My question is quite similar to one in the link: Flux linkage inside of a conductor. But I have trouble understanding any of the answers provided there. Specifically my question is the following: I ...
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Is there a general way of solving the Maxwell equations?

Is there some method for solving differential equations that can be applied to Maxwell equations to always get a solution for the electromagnetic field, even if numerical, regardless of the specifics ...
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What's the physical reason that a massive vector field has only three linearly-independent physical polarizations?

While a four-vector field $A_\mu$ has four components, for a massive field there are only three linearly independent combinations of these components that correspond to physical situations. This ...
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How do you apply “flux-rule” in free space?

Faraday's law of induction states that, the tangential component of the force per unit charge$^*$; EMF produced in a loop of wire, is equal to change in the magnetic flux through the loop, $$\int_C \...
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Electric force on the interface between two dielectrics

Suppose two electrodes maintained at a voltage difference of $V_0$ are separated by a distance of $d=d_1+d_2$. Between the electrodes are two different dielectric medium of $\epsilon_1$ and $\...
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Skin depth; EM wave and AC

When I google skin depth, I get the following definition, Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current ...
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Differences betwen the conformal group and the Schrödinger group?

Facts: The Maxwell (free) equations (4d) are invariant under the 15 dimensional conformal group. The free Schrödinger equation in 3d is invariant under the 15 dimensional group "called" Schrödinger ...
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Alternating current radiates?

Does a wire carrying alternating current release electromagnetic radiation in accordance with Maxwell's equations?
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Amplitude of EM waves

I'm trying to calculate the Amplitude of electric field in the EM waves using the differential forms from maxwells equations. I've been given frequency ($10^8$ Hz) and displacement current density ($...
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Deriving Fresnel Equations for parallel polarization using Maxwell's boundary conditions

I'd first like to preface this post with the "right answer" per wikipedia (I've seen the same answer elsewhere on more reputable websites) The thing I find trouble some is the cross terms such as $...
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Why does gradually increasing refractive index coating reduce reflection?

Why does gradually increasing refractive index coating reduce reflection? EDIT: As @Michael Seifert nicely describes in his answer, reflection only occurs if there is an abrupt change in refractive ...
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How are the two definitions of B, as magnetic field and magnetic flux density, same?

My question, is primarily, based on this question, in which the accepted answer asserts that $\vec B \, $ has two definitions, As magnetic field$^*$, using Lorentz force $\vec F=q \vec v \times \vec ...
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The $z$-component of the electric field vector [closed]

It is asked to find the wave equation for the z-component of the electric field vector by using Maxwell’s equations in free space. Here is my work - In free space we have the following relation ...
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Electromagnetic wave equation: can we ignore the constant of integration?

Suppose we obtain a solution for each of $\mathbf B$, $\mathbf E$ of maxwell equations in the vacuum ($\rho=0$). Clearly, for any constant vector $\mathbf k, \mathbf m$, $\mathbf {B+k}$ and $\mathbf{E+...
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Conformal Invariance of Maxwell's Equations

I am currently doing some conformal field theory (in four dimensions) and want to show the invariance of Maxwell's equations under conformal transformations, in particular \begin{align} \partial_\...
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Light does not always travel on null geodesic?

I am currently reading Wald's General Relativity and a result of section 4.3 stomped me. Part of Maxwell's equation in GR may be written as $$\nabla^a F_{ab} = \nabla^a \nabla_a A_b - R^a_b A_a= 0.$$ ...
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Why does vector potential $\mathbf A$ satisfy $\mathbf E = -\nabla \phi - \frac{\partial \mathbf A}{\partial t}$?

As any magnetic field $\mathbf{B}$ is divergence-free i.e. $$\nabla\cdot \mathbf B = 0,\tag{1}$$ by the solenoidal theorem there exists a vector potential $\mathbf A$ that satisfies $$\mathbf B = \...
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Symmetry of Maxwell equations for electric-magnetic duality

According to Griffiths's book on electrodynamics, including magnetic charge the Maxwell equations become $$ \begin{align*} \nabla \cdot \vec{E} &= \frac{\rho_e}{\epsilon_0} &&& \nabla ...
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What are the differences between the “Time Domain Finite Element” and “The Finite-Difference Time-domain” methods?

Someone knows what are the differences between the "Time Domain Finite Element (FETD)" and the "The Finite-Difference Time-domain (FDTD)" methods for electromagnetics? I want to study transcuteaneous ...
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What in Maxwell's Equations Implies Magnetic Field Lines Concentrate in An Iron?

It's often said in grade school level physics books that an iron will concentrate magnetic field lines. How can this fact be inferred from Maxwell's equations?
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Correlation between the radius of a cylindrical coil with the curl of the electric field induced in that coil

A background of the lab, I am currently working on: I have different sizes of cylindrical coil of wire, each with different radius. I then pass the coil through a near uniform magnetic field and then ...
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What happens to the charge density under parity?

A question came to me when I tried to think about the parity prperties of the Maxwell's equations. The charge density $\rho(\vec{r})$ actually stands for a scalar quantity $\rho(x,y,z)$. Since the ...
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Why all theories are Lorentz invariant?

Ok, in studying of Maxwell equations we have violation of Galilean relativity. This implies necessity of other transformations which make Maxwell equations covariant (invariant in form) under this ...
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Derivation of Hamiltonian in non-linear optics

Update 2: one half of the answer to this question can be found here and the other half in the second half of this post. I asked this question a few hours ago, but decided to re-post it because I ...
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Why the electric field induced in the $\phi$ component, non-zero as compared to the $s$ and $z$ which are zero?

If we take a coil in a uniform magnetic field, move it around to induced some emf around the coil. And so in order to calculate the curl of the electric field induced in cylindrical coordinate system, ...
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Single frequency solutions to wave equation - Two forms? Whats the difference?

I've seen these two forms from multiple sources for solutions to Maxwell's equations: $$ \cos(kz - \omega t) $$ and $$ \cos(\omega t - \vec{k} \cdot \vec{r}) $$ The first one shows a wave travelling ...
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Derivation of Ampere's Law from Biot-Savart

Our aim is to derive $\nabla\times \mathbf B=\mu_0(\mathbf J+\epsilon_0\frac{\partial E}{\partial t})$. To begin with, let $ \mathbf A=\frac{\mu_0}{4\pi}\int_{\mathbb R^3}\frac{\mathbf J(\mathbf r')}{...
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Does the technology exist to generate AC current with a frequency high enough such that when used in a circuit, could generate visible light? [duplicate]

The rapidly fluctuating magnetic field should generate a rapidly fluctuating electric field which propagates away from the wire at the speed of light. I'm not sure how rapidly current can be switched ...
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Derivation: (lossless case) Plane waves have no electric field component in the direction of propagation

Set up Assume that a wave is propagating in the $\hat{z}$ direction with the $E$-field polarized along the $\hat{x}$ direction. In the lossless case, $$0 = \nabla \cdot E = \frac{\partial E_x}{\...
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Are there sources for potential wave solutions for which the electromagnetic far-field is zero?

Are there sources for electromagnetic potential propagating wave solutions for $\phi$ and $\overrightarrow{A}$ for which the electromagnetic far-field $\,>\!\lambda\,$ in vacuum is zero $\...
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Where does the energy go to for a charged sphere with pulsing radius?

In the question A charged sphere with pulsing radius the answer says that the charged sphere does not radiate. However, compressing a sphere of charge to a smaller radius requires work, so where does ...
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Non-relativistic vector first integral

In this paper the author discusses the Poincaré vector $\vec D$, and he wrote the Lorentz equation when magnetic charge is present. Do you have any suggested reading for this and how he derived the ...
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Is there any particular reason why Faraday's and Ampère's laws are valid?

I know that the Maxwell equations are usually the explanation for all electromagnetic phenomena, but I would like to know why those are valid, if there is any reason for them.
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Electric Field in a uniform time-varying Magnetic field

Suppose a homogenous Magnetic Field $\vec{B}$ in vacuum that varies with time, but always points in the z-direction. This induces a curl in the Electric Field $\vec{\nabla} \times \vec{E} = -\frac{\...
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Electromagnetic field of a spinning cylinder

Let us consider an infinite cylinder of axis $(Oz)$ and radius $R$ spinning at a constant radial velocity $\omega$. We assume that this cylinder is made of a metal that is assumed to be a conductor (...
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How to compute the curl of the electric field, experimentally?

In order to experimentally verify Faraday's law and express the curl of the induced electric field, is there any other way to compute the curl without directly working on the cross product (ie working ...
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Investigating the relationship between a changing B field and the curl of the electric field induced

I am currently working on a 12 page lab report on Faraday's law, essentially investigating, how for different magnetic field strength through a stationary loop of wire with N number of turns, effects ...
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Confusion with representing the negative rate of change of magnetic flux in Faraday's law

So Faraday's law states in differential form that $$ \nabla \times \vec{E} = -\frac{\delta H}{\delta t} $$ Using Stoke's theorem, the right hand side (the magnetic flux rate of change) is expressed as ...
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The meaning of the 4-divergence of the 4-magnetic field?

In special and general relativity, the magnetic field is defined as $$B^\mu = F^{*\mu\nu}u_\nu, \label{tag1}\tag{1}$$ where $F^{*\mu\nu} = \frac12 \varepsilon^{\mu\nu\rho\sigma}F_{\sigma\rho}$, and $...
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Magnetic fields and closed loop

It is well known that there don't appear to be magnetic poles. In Maxwell's equations this has the implication $$ \nabla \cdot \mathbf{B} = 0 $$ and results in the statement "the magnetic field forms ...
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How can I show that the speed of light in vacuum is the same in all reference frames?

I have regularly heard that the Michelson-Morley experiment demonstrates that the speed of light is constant in all reference frames. By doing some research I have found that it actually demonstrated ...
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How is it possible for the induced emf to take negative values in Faraday's Law of induction?

Faraday's Law of induction states that the work done per unit charge by the (induced) electric force along a loop of wire, or the emf, is minus the rate of change of magnetic flux through the loop/...
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Intuition of Maxwell's Equations [duplicate]

Is there an intuitive explanation for Maxwell's equations? I know they are axioms but is there a logical understanding of why instead of mathematical. Both forms don't explicate the scientific ...
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Maxwell equation similar to a solution for a standing wave in a box [closed]

From Nature Of Photon: Electromagnetic field The set of Maxwell equations [2] for vacuum is: $$\begin{align} \mathrm{rot} \mathbf{E} &= -∂\mathbf{B}/c∂t, \tag{1} \\ \mathrm{rot} \mathbf{...
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Maxwell equations in curved spacetime with cosmological constant

I am aware of Maxwell Equations in Curved Spacetime. But how do these equations change if the cosmological constant is not assumed to vanish?