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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183 views

Maxwell's equations UPML in FDTD with inhomogeneous media

I'm looking at matching the UPML (uniaxial perfectly matched layer) defined in Taflove&Hagness' Computational electrodynamics to an inhomogeneous media (inhomogeneous w.r.t. both $\varepsilon$ and ...
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$* d * $ operator — Digest the (differential/geometry) meaning

I like to digest better: the $* d * $ operator in Maxwell differential form equation the $* D * $ operator in Yang-Mills differential form equation We already knew that in Maxwell differential ...
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How to show $z$-component of EM plane wave in free space is 0?

I know that for a uniform plane wave propagating in the z-direction in free space, there should be no z-component, however, I am having trouble proving this. Assuming $\vec{E} = E_x (z,t) \hat{x} + ...
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Where am I going wrong in deriving Maxwell's first equation in differential form?

By denoting the source coordinates with prime, I get flux through a closed surface: $$\Phi= \displaystyle\oint_{A} \mathbf{E}(x,y,z) \cdot \mathbf{\hat{n}}\ dA =q (x',y',z')$$ And now using the ...
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Write electromagnetic field tensor in terms of four-vector potential

How can we know that the electromagnetic tensor $F_{\mu\nu}$ can be written in terms of a four-vector potential $A_{\mu}$ as $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$? In the ...
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How did Maxwell figure out the speed of light?

The Wiki article is about 2 graduate years of physics beyond my understanding. What is a good high-school rendition of his thought process: regarding his use of the "distributed capacitance and ...
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111 views

What does Maxwell's equations predict for the propagation of EM waves converging to a point?

Maxwell's equations model EM radiation as propagating away from an accelerating charge. Suppose instead the propagation of this EM radiation is reversed and presented as a source-free boundary ...
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Lagrangian of Phonon-photon

A quite interesting but also hard problem are Polaritons. As far as I have understand the concept it's about phonons coupling to light. The Lagrangian function should therefore have a term for the ...
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Electromotive force in the presence of non-steady currents

Griffiths's Introduction to Electrodynamics states $$\mathcal E = \oint \mathbf f \cdot d\mathbf l$$ In which $$\mathbf f = \mathbf f_s + \mathbf E$$ Where Griffiths describes the summation as ...
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77 views

Electromagnetism and differential forms

I am currently writing a Bachelor's thesis in theoretical physics, and since I like the interplay between mathematics and theoretical physics, I am writing about Maxwell's law in terms of differential ...
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Does a homogeneous oscillating electric field produce a magnetic field?

I am working on a homework problem that says an electron in a continuous laser field can be modeled as experiencing a homogeneous oscillating electric field $\vec{E}(\vec{r},t)=\cos \omega t \ \hat {z}...
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Is the speed of Magnetic field infinite?

My real question is: From Amperes law we know that there is no magnetic field outside of the coaxial cable because the magnetic field generated by the inner wire and outer shell are equal but in ...
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47 views

What is the formula that gives the EMF

I have been recently studying Maxwell Equations, and I wasn't able to understand properly the EMF $\zeta$. Mathematically we have $\zeta=-\dfrac{d\Phi_B}{dt}$ where $\Phi_B$ is the magnetic flux of a ...
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Variational formulation of Maxwell equations with interface/boundary conditions

Consider $\Omega = \Omega_1 \cup \Omega_2$, where $\Omega _1$ and $\Omega_2$ are two different media with conductivity and permeability \begin{equation} \sigma= \begin{cases} \sigma _1 & \text{in ...
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2answers
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Differentiability of electric field due to bounded volume charge distribution

In books on electromagnetism, one often sees expressions of Maxwell's equations like $\nabla \cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$. These expressions make sense if $\mathbf{E}$ (which is ...
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Is tangential component of $\mathbf{B}$ undefined at the boundary of two media?

Tangential component of $\mathbf{B}$ is discontinuous at the boundary of two media. Does this mean that tangential component of $\mathbf{B}$ is undefined at the boundary of two media? If yes, then: $...
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How to connect Maxwell's equations to quantum anomalous Hall effect?

The quantum anomalous Hall effect (QAHE) describes the response of a material resulting from topological properties of its band structure. These topological properties are often characterized by the ...
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Doubt on why magnetic flux density is solenoidal

The $\mathbf{H}$ field can be derived from the potential $\psi$: $$\mathbf{H}=\dfrac{\mu_0}{4 \pi} \int_{V'} \rho \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV' + \dfrac{\mu_0}{4 \pi}...
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A simple proof covariance of Maxwell equations

I read that Maxwell equations are covariant under Lorentz transformations, but I can't find a proof. Or at least a proof understandable by someone that doesn't know higher mathematics (please don't ...
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Static electric field which admits no potential

Conservative condition for (static) electric field $\mathcal{E}$ is usually defined as $\mathcal{E}$ being closed (curl-free). Now this clearly holds when for the given manifold $X$ we have $H_\text{...
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Current sources and conservative electric fields

Suppose a current density ${\bf J}({\bf r},t)$, for which $\nabla \cdot {\bf J} =0$, is compactly supported on a 3D region $R_1$ in vacuum. In general it can produce a nonzero electric field ${\bf E}({...
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An $E$ field with no $B$ field in a region?

Assume there exists a time-varying current source ${\bf J}({\bf r},t)$ in otherwise vacuum and a region $R$ that does not intersect the region of support for ${\bf J}({\bf r},t)$ (That is ${\bf J}({\...
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What causes the magnetic field around a wire? [duplicate]

According to maxwell’s equations, a magnetic field is caused by a changing electric field, but where is this changing electric field in the context of a current carrying conductor that has an induced ...
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Change in electromagnetic field

Suppose you have a configuration with two infinite rods parallel to the $z$ axis, each with uniform charge density $\lambda$. Suppose the rods move. In general, the electromagnetic field they generate ...
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Does the state of curl of the $E$-Field at a point adjust itself instantaneously as soon as $B$ begins changing at a fixed rate, or is there delay? [duplicate]

Faraday’s Law () states that a time-changing magnetic field vector induces a curl of the Electric field around that point. However it does not specify how quickly the necessary spatial gradient is ...
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How can EM waves propagate indefinitely when they really are oscillations of electric and magnetic fields?

I can't claim that I've fully grasped the science behind EM waves, but as far as I understand, EM wave is basically oscillations of electric and magnetic fields kinda switching back'n'forth and ...
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Linearity of Maxwell's equations in tensor formulation

Maxwell equation in tensor formulation are $\partial_\nu F^{\mu \nu}=J^\mu $ and $\partial_{[\gamma} F_{\mu \nu]}=0$. So to show Maxwell equation are linear in vacuum is the following method correct: $...
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How quickly is the desired state of curl of the Electric Field around a point achieved where a Magnetic Field Vector has just begun changing?

My question and context with explanation are given below. Thank you in advance. Faraday’s Law () states that a time-changing magnetic field vector induces a curl of the Electric field around that ...
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2answers
86 views

What are the theorems that constitute the Maxwell's equation? [closed]

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
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A question regarding Faraday's integral law

This might be a quickly phrased question so I apologize if I've missed something trivial in my understanding. Faraday introduced the concept of fields to explain the phenomenon of action at a ...
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30 views

Induced magnetic field in conducting sphere

The last day the teacher solved a problem and I did not follow all of his assumptions. We have a homogeneous conducting sphere in a magnetic field $$\vec{B}=B_0e^{i\omega t}\vec{e_z}$$. This applied ...
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101 views

Can electric displacement field be zero if electric field is not?

The electric displacement field is defined as $$\mathbf{D}=\epsilon_0\mathbf{E}+\mathbf{P}$$ But these equalities hold as well: $$\mathbf{P}=\epsilon_0\chi \mathbf{E}$$ $$\mathbf{D}=\epsilon_0(1+\...
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Where does this form of the induction law come from?

I am currently studying a eddy current disk break and came across this promising paper (PDF, princeton.edu) from 1942. However, in the first formula a suspicious $4\pi$ appears before the current $U(...
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Prove that the electric field produce by a punctual charge is isotropic and radial

I would like to prove mathematically that the electric field produced by a punctual charge is isotropic and radial, i.e. $$\vec{E}(r,\phi,\theta)=E(r)\vec{e}_r\tag{1}$$ I think that this statement ...
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Simple explanation to the induction from the slowly changing $\vec B$ of a solenoid in the region of $0$ magnetic field

I would like to get some elementary intuition into the problem a solenoid fed with a time-dependent current, and the resulting current that such the solenoid field would induce in a loop completely ...
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1answer
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How to derive $c=1/\sqrt{\varepsilon_0\,\mu_0}$ from integral form of Maxwell equations? [closed]

I've read similar questions and answers given thereto but find them unsatisfactory. So please don't mark my question as "duplicate". The question may as well be a duplicate, but it's still waiting for ...
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How to choose the boundary condition for Maxwell's equations in the vacuum?

I need to solve the Maxwell's equations with sources in the vacuum numerically. The simplified problem is as following. A charged particle moving along the $z$ direction with speed $v_z$. Then, it ...
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How are the differential forms for Maxwell's Equations used?

I am currently reading up on Maxwell's Equations (specifically Ampere's Circuital Law- with Maxwell's Addition) for a presentation on differential equations. I chose the topic ignorant of how the ...
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1answer
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Does superposition of all possible plane waves represent complete solution of Maxwell's equations in free space?

Consider the set of all possible superpositions of all possible "plane waves that satisfy Maxwell's equations in free space". Does this set represent all possible solutions of Maxwell's equations in ...
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Polarization on a spherical electromagnetic wave in free space using classical electromagnetism

The polarization of a plane wave traveling in free space is well defined and traverse to the direction of propagation from classical electromagnetic theory. Spherical waves are another type of ...
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3answers
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Solution to Maxwell's equations in free space that are not plane waves [closed]

Are there solution to Maxwell's equations in free space that are not plane waves? I think there aren't. (Save trivial ones, i.e. E=const , B=const ) But i am not able to prove it. Please help. I would ...
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Derivations of Maxwell equations

In my book of electrodynamics, the Maxwell equations are always used for specific conditions (electrostatics, magnetostatics, …). But nowhere I see a complete derivation of the equations. Maybe it ...
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How to extract all information regarding electric and magnetic fields from Maxwell's equations in free space? [closed]

Is it possible to obtain some set of equations in E and B that have decoupled E and B , but are exactly equivalent to Maxwell's equations in free space? Exactly equivalent in the sense that the any ...
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Maxwell's equation in free space from wave equations of electric and magnetic field

How to go from the wave equations of electric and magnetic field and $$ \boldsymbol{\nabla}\cdot \mathbf E = 0 \quad \text{ and } \quad \ 0 = \boldsymbol{\nabla}\cdot\mathbf B, $$ to the remaining ...
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1answer
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Flux of Electric field non-zero through a surface for light in free space?

Consider an electromagnetic wave propagating through free space.The Electric and magnetic components of the fields, vary as sinusoids. If I construct a sphere of radius $\lambda/4$ at any location; i ...
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Are wave equations equivalent to Maxwell's equations in free space?

In free space, do Maxwell's equations contain the same amount of information regarding electric and magnetic fields as is contained in the wave equations derived from them? If so, how?
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2answers
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How do we get a expression for speed of light $\frac{1}{\sqrt{\mu\epsilon}}$ using Maxwell's equation? [duplicate]

I am a $12^{th}$ class student in India. I am quite new to these interesting concepts. And, I think I've learnt about electrostatics, magnetism, Maxwell's equations very thoroughly. But, my books ...
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Maxwell stress tensor for electromagnetic wave

Sorry if this is a naive question but I've been struggling in trying to proof this for a week. Consider an electromagnetic wave with wave vector $\vec{k}=k\hat{n}$, the Maxwell stress tensor can be ...
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How can we prove the law of Ohm $V=I*R $ from Maxwell's equations [duplicate]

$$∇⋅E=ρϵ0$$ $$∇⋅B=0$$ $$∇×E=−∂B/∂t$$ $$∇×B=μ0(J+ϵ0 ∂E/∂t)$$
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Feynman-Heaviside formula and Mach's principle

I was wondering if the Feynman-Heaviside formula for the electric field of a moving charge could be used to write down the force/reaction force between charges $q_1$ and $q_2$ in a Machian purely ...