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 can one $E$-field formula be rewritten to another version [closed]

I will start with going through a rewriting of one E field to another that I understand which is that the total E field out of a sphere is derived as $$\pmb{r}=x\pmb{i}+y\pmb{j}+z\pmb{k}=r\cos\theta \...
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Quasi-classical model of an atom

This question does not have a close relationship to this or this Stack Exchange question. It's not about the Bohr atom model nor the Rutherford atom model, but is fairly closely related to this3 SE ...
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105 views

Electromagnetic fields and electromagnetic wave boundaries

I have first learnt the concept of plane electromagnetic waves as a solution to Maxwell's equations in vacuum recently and they are usually depicted them as follows: Here is my confusion: in the ...
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Separating static and time-dependent components of E and B

I have been wrestling with Maxwell's equations using a source term in which both charge density and current density are time-independent functions of position, and where current is considered to be ...
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The amplitude for the electric field of a particular electromagnetic plane is given in the direction $\mathbf{\hat{j}}$?

Example 2.6 of Optics, Fifth Edition, by Hecht, is as follows: We will see in the next chapter, electric field ($\mathbf{E}$) of a particular electromagnetic plane wave can be given by the expression ...
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Prove that Maxwell Equations are Lorentz covariant - Continued [closed]

I know that there are already half a dozen threads asking and answering my question, but this is not a duplicate. I found many answers, but the most helpful to me is the following: Proof that Maxwell ...
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108 views

Varying Potential Difference for Displacement Current

Displacement current is the result time varying electric field between two plates of a capacitor and is constant when rate of change in potential difference is constant. Now the question is whether ...
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How does the EM-field really look like in the foci?

In geometric optics, the ancient Greeks and Descartes have learned us that... ...a parabolic mirror reflects lightrays coming in from infinity all towards its focus. ...an ellipsoidal mirror (with ...
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Is the E field generally conservative outside the cases when vector potential goes to zero?

In Ronald Wangsness' "Electromagnetic Fields" book, he states that $\textbf{E}$ generally depends on both a scalar and vector potential in the form, $$\textbf{E} = -{\bf \nabla}\Phi -\frac{\partial \...
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Confused about Feynman Lectures on attraction between two charges or wire as relativistic phenomenon

I am confused on one of the earlier sections of Feynman Lectures Vol 2. I've posted the relevant snippet below. Since it's in the public domain (http://www.feynmanlectures.caltech.edu/) I think I can ...
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Volume required to calculate D and H macroscopic variables

For a dielectric, we replace E, B by D, M realizing that the latter pair is a macroscopic property averaged out over a large number of molecules. What is the lower bound on number of molecules or ...
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What shape does the Induced Electric Field Lines take when in a region there is Non Cylindrical Time Varying Magnetic Field?

Everywhere in the literature I have across, I find that whenever the discussion of the Induced electric field comes, it is discussed around a cylindrical time-varying magnetic field. The induced ...
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Maxwell and Newton [duplicate]

Has anyone tried to modify Maxwell's electromagnetic field theory so that it is invariant under the Galilean transformation?
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Rays and the ray equation: behaviour in optical systems, valid solutions to Maxwell's equations

In my studies of geometrical optics, I have encountered variations of the claim that, from what I recall, when the amplitude of the wave is changing slowly, the ray equation is a valid solution to ...
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Number of independent components of the electromagnetic field

I'm taking a course on electrodynamics and I'm confused when we start talking about potentials. The electromagnetic field seems to have 6 independent components to me. It's described by six dynamical ...
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316 views

Curl of electric field

Related: How is the curl of the electric field possible? According to my information: $$\nabla \times \frac{\hat{r}}{{|\mathbf{r}|}^2} = 0 $$ for all $\mathbf{r}$ ( including $\mathbf{r}=0$; ...
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Sources of Electromagnetic Potentials

I have a question regarding sources of electromagnetic potentials: As far as I know, wave equations for the scalar and magnetic vector potentials can be derived from Maxwell equations employing the ...
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How is the surface integral of $\vec J$ carried out in Ampere's law?

In Ampere's law, the right hand side consists of the surface integral $\int_s\vec j\cdot ds$ which is supposed to represent the current passing through the surface of the enclosed volume. However, $\...
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Applying Ampere's law in situation with non-physical $E$-field?

On an exam I was given this question: Suppose an electric field in a region with no current $(\textbf{J}=\textbf{0}$) is given by $\textbf{E}(t,x,y,z) = \sin(\omega t)\hat{\textbf{k}}$ and $C$ is ...
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Which Maxwell equation explains how electric engines work

While I understand that electric generators (magnetic field $\to$ electric current) work according to Maxwell-Faraday law, some people say that electric engines (electric current $\to$ magnetic field) ...
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Are the boundary conditions purely a consequence of Maxwell's equations?

The boundary conditions, namely were all these, realized only by looking at Maxwell's equations? Or is there a physical reasoning behind them? For example, Why does the component of the electric ...
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From Maxwell equations to wave equation: what is the meaning of the differentiation?

Starting from the Maxwell's equation in empty space and without charges and currents, we can write the two following equations: $\bar \nabla \times \bar E= -{{\partial \bar B}\over{\partial t}}$ $\...
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Time it takes a coin to fall in a magnetic field

This was a homework problem of mine was: "A coin is placed on its edge in a vertical magnetic field of $B=20$ T. The initial angle between the coin and vertical is $\theta_0=0.1$ rad. Estimate how ...
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Topological term in Lagrangian of Maxwell's theory

I should derive the equation of motion for the case, when the Lagrangian of Maxwell's theory has the additional term $\Delta \mathcal{L}=cF_{\mu\nu}\widetilde{F}^{\mu\nu}$ with the dual tensor $\...
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How does the concept of plane wave make sense?

The speed of light can be used by applying Maxwell's equation to plane waves and EM wave equation is found using the plane wave model and the wave equation. But if we think of what is actually ...
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Faraday Law: integral and differential forms

Let's consider both the integral and differential equations which express the Faraday Law (3rd Maxwell Equation): $$\oint_{\partial \Sigma} \mathbf E \cdot d\mathbf l = -\frac{d}{dt}\iint_{\Sigma} \...
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Two different forms of the electromagnetic wave equation

Wikipedia says that the electromagnetic wave equation written in terms of the electric field $\mathbf{E}$ is $${\displaystyle {\begin{aligned}\left(v_{ph}^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\...
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How does one check whether an equation is Lorentz Invariant or Galilean Invariant?

As a physics student, I hear this term a lot that this equation is Lorentz Invariant or galilean Invariant e.g Dirac equation is Lorentz Invariant. Even in a non-linear pde class e.g the KdV equation ...
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Relation between $\textbf{E}$ and $\textbf{H}$ field for TM wave

My book (Nanoscale Energy Transport & Conversion by Gang Chen, pg. 173) considers the following scenario of a TE wave reflecting at an interface: It says the incident $\bf{E}$ field is: $$\textbf{...
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Foundations of Maxwell equations

Suppose we have two particles with charge $q_1$, $q_2$ with velocity $v_1$,$v_2$ in positions $r_1$, $r_2$. We know that the particle 2 generates an electric field and a magnetic field and in position ...
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What's wrong about Maxwell's equations for a Hall probe?

I am using Maxwell's equation to analyse the current and electric field of a Hall Probe. A Hall probe is basically a thin sheet of metal with a current through it. When a uniform magnetic field $\def\...
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What's the solution of Maxwell's equation to the electromagnetic field in circular cylindrical conductors? [closed]

In the image below there are the first steps in a book on power cables to calculate the distribution of current in a conductor due to skin and proximity effect. I don't understand how the author used ...
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Conceptual Understanding of Zero Curl in Ampere's Law

I understand that Ampere's law tells us that the current density times $\mu_0$ at some location must be equal to the curl of $\mathbf{B}$ at that location. However, conceptually this is troubling me. ...
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Force per unit area on a solenoid

I am having a hard time attempting to eliminate constants of integration while solving maxwell's equations. Here is an example of a solenoid. Let a wire carrying current $I$ wind $n$ times per unit ...
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How to show the that Lorenz gauge is true given that the scalar and the magnetic vector potentials are not unique?

So I understand that we select the divergence of A (magnetic vector potential) to be: $$\frac{1}{c^2}\dot{\phi} + \nabla\cdot\vec{A} = 0.$$ The Lorenz gauge (1). because of the symmetries in ...
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How is $G_0 = f(t \pm r/c)/r$ a solution to Poisson's Equation For A Point Source?

In his 1972 book, "The Classical Electromagnetic Field", on p.184, Leonard Eyges States, I don't see how Equation (11.41) is a solution to (11.40)
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“Derivation” of continuity equation

The surface integral of j over a surface S, followed by an integral over the time duration t1 to t2, gives the total amount of charge flowing through the surface in that time (t2 − t1): $${\...
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What unit system is used in Feynman's Lectures on Physics, Pt. 2

In part II, lecture 18, of Feynman's Lectures on Physics, on Table 18-1 Feynman writes Ampère's law as $$ c^2 \nabla \times \vec{B} = \frac{\vec{j}}{\epsilon_0} + \frac{\partial \vec{E}}{\partial t}. $...
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How magnetic monopole modifies faraday's law

I have encountered a statement that upon existence of magnetic monopoles, the equation $$\nabla \times \textbf{E} = -\partial_t \textbf{B}$$ must be modified. I only see that the last equation $$\...
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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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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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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 between 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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