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 Maxwell-Boltzmann distribution, or the thermodynamical equations known as Maxwell's relations.

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Is there a space associated with a collection of functions which holds all consequences of said functions as objects?

I am an undergraduate student studying physics. I am working through a linear algebra book this summer to deepen my understanding, so I am thinking generally about organizing mathematical objects. I ...
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Observing attraction between parallel wires from two perspectives

Two parallel wires through which there flows a current in the same direction. While the electrons are moving, obviously the wires attract each other (the right hand rule for magnetism and electricity)....
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Can we derive Ampere's Circuital Law from Gauss's Law or vice versa?

I was curious if it is possible to derive Ampere's Circuital Law from Gauss's Law as they are very similar and both can be applied for highly symmetrical problems $(Infinite\space wires,Rings..etc)$ ...
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Why are the electric and magnetic fields of a charged particle a single entity and should be mutually orthogonal?

I think Maxwell's four well-known equations dictate the correct answer. I guess the question is a general rule (no specific problem) in electromagnetism as well as Maxwell's 4 equations themselves. ...
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Are scalar and vector potentials solutions to Maxwell's equations? [closed]

Are scalar and vector potentials $\phi = 0$, $\vec{A} = A_0 \cos (kx-\omega t) \hat{e}_z$ solutions to Maxwell's equations? Do these potentials correspond to the Lorentz and Coulomb calibration?
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Does the electric field due to Faraday's law have zero divergence? [duplicate]

How do we know that the electric field that is produced due to Faraday's law of induction does have zero divergence since when proving Maxwell's equation of gauss law, where the divergence of electric ...
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How should I use the complex permittivity of a material?

Here: $$\epsilon = \epsilon' + j *\epsilon''$$ I understand that the first part ($\epsilon'$) is the relative permittivity of a material, while the second part $\epsilon'' = \frac{\sigma}{\epsilon_0\...
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Can the electric field have closed field lines?

In electrostatics, we know that $\vec{\nabla}\times\vec{E} = 0$ and so, the field lines can't form loops. But when we have time-dependant magnetic fields, there's the Faraday-Lenz law which tells us ...
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How to calculate the field distribution due to a current radiating in one direction?

Given a surface current distribution, $\textbf{J}(r)$, we can calculate the magnetic potential in the Lorenz gauge using $$ \textbf{A}(\textbf{r})=-μ_{0} ∬g(\textbf{r},\textbf{r}')\textbf{J}(\textbf{r}...
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Gravity Differential Equations

I was just messing around with Newton's Law of gravitation, when I had the idea of converting Newton's Law into differential form (more or less like Maxwell's equations). I did the following: #1 ...
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Confusion in the showing EM wave exist from Maxwell equation

When deriving the mathematical description of a field, we set the current density and charge to zero in Maxwell's equations. However, this condition is not absolutely true anywhere on earth. Yet, we ...
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Can the Lorentz force equation in curved spacetime be derived from the Einstein-Maxwell equations?

Given the Einstein field equations, $$R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} = \kappa T_{\mu\nu}$$ that imply in particular that $\nabla_\mu T^{\mu\nu}=0$, one can show, using the explicit form of $T^{\...
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Example of Maxwell Equations with 24 boundary conditions

In Christopher Bairds supplementary notes on the Uniqueness of Maxwell Equations, he says that in full generality one needs 24 boundary conditions to uniquely determine a solution of Maxwells ...
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Do Maxwell's equations still apply in fluids?

Hi I am trying to build an electrical impedance tomography platform. The problem I'm facing mainly is to come up with an algorithm to calculate the variation of conductivity within the lung tissue ...
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Ohms law validity and EM waves in conductors

Ohms law, when $\sigma$ is real: $$\vec{J} = \sigma \vec{E}$$ Is derived from the steady state solution of the equation: $$m\frac{d\vec{v}}{dt} = e\vec{E} - \frac{m}{\tau}\vec{v}$$ Where $\vec{E}$ has ...
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Retarded potential of a point charge

Potential of a moving point charge is given as $V (\mathbf r,t)= \frac{1}{4\pi\epsilon_0}\int \frac{\rho (\mathbf r',t_r) }{|\mathbf{ (r-r')}|}d\tau'$ Griffiths says: " It is true that for a ...
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Possible error in Griffiths, Intro. to Electrodynamics, Problem 7.60

Probelm 7.60 of Griffiths' Introduction to Electrodynamics, 4th ed, says: Suppose $\mathbf J(\mathbf r)$ is constant in time but $\rho(\mathbf r, t)$ is not—conditions that might prevail, for ...
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Expressing Maxwell's equations in tensor notation

I've been teaching myself relativity by reading Sean Carroll's intro to General Relativity textbook, and in the first chapter he discusses special relativity and introduces the concept of tensors, ...
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Capacitance of caapcitor with complex dielectric material

How do I find capacitance of capacitor if we insert a material in capacitor that has a dielectric constant which depends on position inside a capacitor? For example two plates of capacitor filled with ...
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Why is the wave impedance calculated as $\eta = \eta_0 / \sqrt{\epsilon_r} = \dfrac{377}{\sqrt{4.0}} = 188.5 \ \Omega$?

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following: Plane Waves in a Lossless ...
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Derivation of $~\nabla^2\mathbf{H}=\sigma\mu{\partial\mathbf{H}\over\partial\mathrm{t}}+\epsilon\mu{\partial^2\mathbf{H}\over\partial\mathrm{t^2}}~$

The following equation(subset of Maxwell's equations of electromagnetic wave(s)) is said held in free space. $$\underbrace{ \color{fuchsia}{\nabla^2\mathbf{H}_{}=\sigma\mu{\partial\mathbf{H}_{}\over\...
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Interference of a current carrying conductor on other cable

I need to solve a problem. Below figure explains the setup. We have a current carrying conductor (Wire A) which is carrying 3A of current, which is generated by a 400 kHz sine wave source We have ...
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London equations and flux pinning

I am having some conceptual issues with the London equations, namely the equation for the magnetic field and penetration depth. $$ \frac{d\vec{J}}{dt} = \frac{ne^2}{m}\vec{E}$$ Behavior of magnetic ...
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What exactly are relativistic fields?

Is magnetic field relativistic? Is electric field relativistic? How do you imagine something causing an actual physical difference to be purely dependent on the choice of a reference frame?
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Using "Maxwell's curl equations" to get $H_y = \dfrac{j}{\omega \mu} \dfrac{\partial{E_x}}{\partial{z}} = \dfrac{1}{\eta}(E^+ e^{-jkz} - E^- e^{jkz})$

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following: The Helmholtz Equation In ...
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How does $-k^2\left( E^+ e^{-jkz} + E^- e^{jkz} - E_x \right) = 0$?

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following: Plane Waves in a Lossless ...
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Charge travelling with constant speed and Maxwell equations

I am reading "Matter and Interaction" by Chabay and Sherwood chapter 23 (4 edition). They state that accelerating charge generates wave satisfying Maxwell equations. They also provide an ...
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Reduction of the Helmholtz equation for an electric field with only an $\hat{x}$ component and uniform (no variation) in the $x$ and $y$ directions

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following: The Helmholtz Equation In ...
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Coulomb gauge with $\rho = 0$ implies Lorenz gauge?

Maxwell equations take the form: $$\nabla^2 \phi + \frac{\partial}{\partial t} \nabla \cdot \vec{A}= - \frac{\rho}{\epsilon_0}\qquad (\nabla^2 \vec{A} - \mu_0\epsilon_0\frac{\partial^2 \vec{A}}{\...
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Maxwell's stress tensor and pressure

I am studying Electromagnetism from Griffiths and in the book it is stated that diagonal elements of Maxwell's tensor represent pressure. I want to calculate pressure on the wirings of an infinitely ...
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Helmholtz decomposition of $\vec{E}$ field

Edit: Can someone check my answer and possibly complete my task at the end? The helmholtz theorem states that any vector field can be decomposed into a purely divergent part, and a purely solenoidal ...
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Maxwell's eq-meaning of del's cross and dot product?

In maxwell's eq there is del whose cross and dot products exist. So what is del in cross vs dot product. What's the difference when it's just a partial differential operator.
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Disagreement in the magnetic field inside a capacitor depending on the Maxwell equation

Consider a capacitor of two parallel circular plates of radius $a$ separated by an air gap of width $d$, where $d \ll a$. The capacitor is connected an alternating voltage source, $ V = V(t)$. To find ...
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Understanding how changing current relates to induced EMF for an inductor

When deriving the voltage drop across an inductor, we can consider a loop as shown that starts at terminal $a$ and goes through the coil to terminal $b$ before returning to terminal $a$ outside (in a ...
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How to deduce the structure of an EM wave [closed]

Suppose we have, in vacuum, the following electric field: $$\vec{E}(z,t)=\sqrt{\frac{2\omega ^2}{\varepsilon _0 V}}q(t)\sin\left(kz\right)\hat{x} \tag{1}$$ in my lecture notes it is stated that then ...
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Determinating the position and charges of particles from electric potential

Special configuration of charges gives us electrical field with scalar potential described by equation $$U(\vec{r})=q\ln\sqrt{\frac{r-\vec{a}\cdot\vec{r}}{r+\vec{a}\cdot\vec{r}}}$$ where $q$ is ...
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Because Maxwell's equations are not all linearly independent, the six boundary conditions in the above equations are not all linearly independent

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a Dielectric Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says the ...
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Why is the tangential component of electric field is equal on both sides of interface?

From Maxwell's equations (gauss's law) : $ \hat{n} \cdot (\vec{D_2}-\vec{D_1})= \sigma \implies D_{2\perp}-D_{1\perp}=\sigma : $ surface charge density. How does one go on proving further that $E_{1\...
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Is this "Transfer" correct?

For single div--curl system, i.e. $$\nabla\cdot {\bf u} = f,\quad \nabla\times {\bf u} = {\bf b}, \tag1$$ the theorem 3.5 in this paper ( Junichi Aramaki, L^p Theory for the div-curl System, Int. ...
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How does QED explain the circular magnetic field around a straight conductor?

Since Maxwell's equations are phenomenological, I'm looking for the actual deep reason why the magnetic field is oriented circularly around a straight conductor. Could anybody knowledgeable on quantum ...
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What scale are Maxwell's Equations valid?

Both Maxwell's equations and quantum mechanics are used to describe the behavior of electrons in circuits. I am confused on the interlinking between the two and the dividing line between when you use ...
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Why do we have the negative on the RHS of $\Delta \mathscr{l} E_{t1} - \Delta \mathscr{l} E_{t2} = -\Delta \mathscr{l} M_s$?

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a General Material Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says ...
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What is the exact calculation that leads to the phasor form term $-j\omega \int_S \bar{B} \cdot d\bar{s}$?

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a General Material Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says ...
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What is this difference in the notation for the electric flux density in this closed surface example?

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a General Material Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says ...
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Wireless Power Transfer in Layered Material

I’m looking to develop a mathematical model for near-field electromagnetic power transfer through layers of human tissue, from an external transmitting antenna to an embedded antenna. Each tissue ...
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Aren't these equations identical? What is the difference in the aforementioned "time-harmonic version"?

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a General Material Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says ...
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How exactly does an inductor resist change in current?

To my understanding, the electric field inside an inductor is zero if the inductor is made of an ideal wire. According to this post, this happens because the induced field is canceled out by the ...
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Iridescent duck and photonics in nature. Why duck's head is only green?

We all know and adore pure beauty of mallard male: I know that such iridescent colours are usually a result of feathers forming a photonic crystal with layers of air sandwiched between layers of ...
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Making Lorentz equation dimensionless for simulations

\begin{equation} \frac{dv}{dt}=\frac{e}{m}(E+\frac{v}{c}\times B) \end{equation} I'm making a numerical simulation and I need to make the equation dimensionless. I'm having trouble in doing so. I was ...
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What happens when you put a conductor inside a cavity, contaning a $B$ field, of another conductor?

My problem is this: A perfectly conducting material has a single cavity carved out inside it. The $B$ field in the cavity has the form $$B(r,t) = \text{Re } B_c(r) e^{-i\omega t}$$ and I've already ...
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