The four fundamental fundamental equations of electromagnetism.

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17
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Deriving the speed of the propagation of a change in the Electromagnetic Field from Maxwell's Equations

I've been told that, from Maxwell's equations, one can find that the propagation of change in the Electromagnetic Field travels at a speed $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ (the values of which can ...
3
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1answer
89 views

How to solve “EM wave equation” for the field of uniformly moving charge?

Is it possible to show that the field of a uniformly moving charge, which is according to Biot-Savart law is given by... $${\bf E}({\bf r},t)=kq\left(\frac{1-v^2/c^2}{(1-v^2 \sin^2 ...
0
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0answers
12 views

Derivation of Poynting theorem in matter

In most textbooks I have read they derive the Poynting theorem using the Maxwell's Equation in vacuum and the fact that the force density $f=\pmb{E} \cdot \pmb{J}$. Then they just generalize it ...
6
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1answer
375 views

Biot-Savart law from Ampère's with multivariate calculus

Let us assume the validity of Ampère's circuital law $$\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}$$where $\mathbf{B}$ is the magnetic field, $\gamma$ a closed path linking the ...
5
votes
3answers
1k views

How is the curl of the electric field possible?

Taking the curl of the electric field must be possible, because Faraday's law involves it: $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$ But I've just looked on Wikipedia, where it ...
0
votes
1answer
299 views

Surely force on shell can't be balanced by field momentum?

Imagine a particle with charge $q$ at rest at the origin. It is surrounded by a concentric spherical insulating shell, also at rest, with charge $Q$ and radius $R$. At time $t=0$ I apply a constant ...
0
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3answers
644 views

How to derive the expression for the electric field in terms of the potential?

How can I derive that $$\vec{E}=-\vec{\nabla}\phi-\frac{\partial \vec{A}}{\partial t}$$ where $\phi$ is the scalar potential and $\vec{A}$ the vector potential?
0
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0answers
10 views

Why does a 2-sided propagating EM wave become 1-sided if B is made proportional to E?

If you simulate the propagation of an electromagnetic wave in 1D free space (no charges or currents) with initial conditions of $E\neq0$ and $B=0$, and you look at a movie of $E$ vs time, then after ...
0
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0answers
26 views

Problem with understanding boundary conditions in electromagnetism

In some books on electrodynamics they stress that electric current won't radiate if it is placed on a perfect electrical conductor (PEC), citing image theory: exactly opposite current will appear and ...
0
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1answer
37 views

Relation between displacement current, dielectric and time varying Electric field

I know that displacement current is produced in dielectric material due to dipole moment. I also know that displacement current is produced by time varying electric field (according to maxwell ...
1
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3answers
172 views

Maxwell's equations - underdetermined - uniqueness

Maxwell's equations can be seen as two dynamical equations (the two curl equations), and two constraint equations (the two divergence equations). So we have 6 unknowns ($E_x,E_y,E_z,B_x,B_y,B_z$). ...
45
votes
7answers
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Do Maxwell's Equations overdetermine the electric and magnetic fields?

Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\vec{E}=(E_x,E_y,E_z)$ and ...
2
votes
2answers
210 views

Boundary conditions for Maxwell's equations at the interface between two media

Consider the following simple Maxwell's equations: $$ \nabla\cdot\mathrm{D}=\rho $$ $$ \nabla\times\mathrm{E}+i\omega\mathrm{B}=0 $$ $$ \nabla\cdot\mathrm{B}=0 $$ $$ ...
-1
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0answers
31 views

How can there be E and B fields inside waveguides?

A hollow waveguide that is made of conducting material allows for the existence of an electromagnetic wave inside it which propagates from one end to the other. However, conductors are known to ...
2
votes
1answer
51 views

What causes electromagnetic waves to propagate in free space?

In free space, $\rho=0$ and $J=0$, so there are no electromagnetic sources/sinks. Maxwell's equations thus reduce to: $\nabla\cdot E = 0$ $\nabla\cdot B = 0$ $\nabla\times E = -\frac{\partial ...
0
votes
1answer
397 views

What are the boundary conditions for EM waves normally incident on the interface between two dielectric media?

An EM wave, amplitude $E_0$, frequency $\omega_0$, is incident upon a material with relative permittivity (dielectric function) $$\varepsilon \left( z \right) = \left\{ \begin{gathered}{\varepsilon ...
0
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1answer
54 views

Derivatives involving four vectors [closed]

The Schrödinger lagrangian for complex fields is $$L=\frac{1}{2m}(D_i \psi)^* Di \psi - \frac{i}{2} \left[\psi ^* D_0 \psi - (D_o \psi)^* \right] - \frac{1}{4}F^{\mu \nu}F_{\mu \nu}$$ Where ...
0
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0answers
21 views

Which formulas would tell me the gradient of an electromagnetic field at an arbitrary distance from a pole? [duplicate]

I'm a newbie to physics and was wondering where I can read about electromagnetic gradients. From what I understand (and my intuition) electromagnetic fields create force gradients around its poles. ...
1
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1answer
150 views

Spherical magnet inside a solenoid

When passing a bar magnet through a long solenoid why is it that the induced emf when the magnet is in the middle of the solenoid is zero? And if a spherical magnet is put inside the solenoid, will ...
0
votes
1answer
145 views

How can electric field representation be obtained from Enge representation using Maxwell's equations?

Suppose we have a long electric capacitor. Let $L$ be its length ($z$ coordinate), $W$ its width ($y$ coordinate), and $D$ its full height (full aperture; $x$ coordinate). Let $L\gg W\gg D$. The ...
13
votes
4answers
2k views

What could magnetic monopoles do that electrically charged particles can't?

I understand the significance to physics, but what can a magnetic monopole be used for assuming we could free them from spin ice and put them to work? What would be a magnetic version of electricity? ...
1
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1answer
48 views

Questions about Biot-Savart law and Ampere's law

A textbook I'm studying with described finding vector magnetic potential $\vec{\text{A}}$ from Biot-Savart law as below. ...
0
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0answers
19 views

wave propagation modelling

what is the best modelling technique for modelling mm-wave propagation in electromagnetic environment. Right now,am working on how to use use Transmission-line matrix (TLM) and ray-tracing techniques
1
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1answer
49 views

In a waveguide, where does the energy in attenuated waves go?

In an electromagnetic waveguide, there is generally a "cutoff frequency." Electromagnetic waves with a frequency that is lower than this cutoff frequency will not propagate at all -- i.e., they will ...
4
votes
2answers
77 views

Maxwell's equations from differential forms

I found the following in some lecture notes I took some time ago: $$ \mathbf{E}=-\text{grad}\Phi-\partial_t\mathbf{A}\\ \mathbf{B}=\mathrm{rot}\mathbf{A} $$ These are the electromagnetic fields ...
5
votes
2answers
440 views

Proof of Ampère's law from the Biot-Savart law for tridimensional current distributions

Let us assume the validity of the Biot-Savart law for a tridimensional distribution of current:$$\mathbf{B}(\mathbf{x})=\frac{\mu_0}{4\pi}\int_V ...
4
votes
2answers
144 views

Missing Hypothesis in Electromagnetism Texts

In the Feynman Lectures, Chapter 21, I find the statement We have solved Maxwell's equations. Given the currents and charges in any circumstance, we can find the potentials directly from these ...
1
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1answer
41 views

Is Gauss law still true in dielectric material?

In vacuum we have $$\nabla \cdot \mathbf{E} = \frac {\rho}{\varepsilon_0}.$$ Can we still use this formula when there's dielectric material in space? Where $\rho$ is total charge density.
0
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0answers
39 views

Displacement current in a capacitor

I recently learned about the concept of displacement current in a capacitor. I've understood the basics - mainly understood the reason why it was introduced. However, is it purely a mathematical ...
1
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0answers
105 views

Aether/Ether in Electrodynamics? [closed]

From reading Maxwell's original papers on the formulation on the dynamics of electromagnetism, I saw that Maxwell kept mentioning aether in his paper. I know today in modern physics the Michelson ...
0
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1answer
26 views

Does the magnetic flux spin in the same direction when the electric field is growing and collapsing?

I have three concentric iron pipes, stacked inside each other. It looks like a bulls-eye when viewed from above. To the innermost and outermost pipes I have connected a battery through a switch. ...
1
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1answer
56 views

Can a charge moving in an open trajectory qualify as current?

It is sometimes said that a point charge is equivalent to an electric current. If it were a steady current, I should be able to find it from Ampere’s law or Biot-Savart’s law. Even if the current is ...
2
votes
3answers
90 views

Meaning of the Vector Wave Equation

So I thought I would try my luck here on physics stack exchange about an intuitive meaning of the Vector Wave Equation. I know there are a lot of resources out there that explain this equation, but ...
0
votes
2answers
586 views

Electric field from current without Maxwell's law of induction

A long, straight wire carries a current that decreases linearly with time. What is the direction of the induced electric field outside the wire? I would interpret this as follows: a current ...
5
votes
4answers
345 views

The necessity of the B field

It is fairly easy using basic special relativity to arrive at the conclusion that the magnetic force effect on nearby charges of wires carrying currents on nearby charges is only due to the length ...
1
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2answers
47 views

Derivation of Displacement current term in Ampere's Law

I have a quick question: In deriving the displacement current term for Ampere's Law, my book has the line: $$\Phi_E= \int_S \mathbb{E} \cdot \hat{n} da= \int_S \frac{\sigma}{\epsilon_0} da = ...
6
votes
2answers
7k views

Why is the divergence of a magnetic field equal to zero?

We know due to Maxwell's equations that: $$\vec{\nabla} \cdot \vec{B}=0$$ But if we get far from the magnetic field, shouldn't it be weaker? Shouldn't the divergence of the field be positive? If ...
-1
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1answer
263 views

Application of Displacement Current

I'm reasonably happy with the derivation and results of displacement current, however, I'd like to be aware of a few practical applications of this idea. So far, the only one I'm aware of is when ...
5
votes
4answers
2k views

can one introduce magnetic monopoles without Dirac strings?

To introduce magnetic monopoles in Maxwell equations, Dirac uses special strings, that are singularities in space, allowing potentials to be gauge potentials. A consequence of this is the quantization ...
3
votes
1answer
147 views

How is Biot-Savart law verification of Maxwell's 4th equation for steady current?

Please provide some theoretical procedure which equates Biot-Savart law with the Maxwell's 4th equation for steady current, which is Ampere's law $$\quad \nabla\times{\bf B} = \mu_0{\bf J}.$$
9
votes
4answers
584 views

Do the integral forms of Maxwell's Equations have limited applicability because of retardation?

In the usual bookwork treatment, it is easy to show that the differential and integral forms of Maxwell's equations are equivalent using Gauss's and Stokes's theorems. I have always thought that ...
2
votes
1answer
454 views

The truest/most general Maxwell's equations in isotropic, linear, inhomogeneous media with sources

Sources use $\mu H=B$ and $\epsilon E= D$, assuming homogeneous media. Obviously if $\mu$ is space varying, $\nabla . (\mu H)$ need not be equal to $\nabla . B$ What is the most general form for ...
0
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0answers
46 views

Does Gauss flux theorem hold in relativity?

Does the Gauss flux theorem, stated in the classical electrostatics as $\iint{\vec{E}}\cdot{\vec{dS}}=q/\epsilon_0,$ hold in the case of relative motions. For instance if we observe a charged body ...
1
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0answers
16 views

Relating noise spectral densities of magnetic flux and voltage

My problem is as follows. I generate a voltage $V$. This voltage is applied to a resistor R, producing a current $I$. This current is then passed through a superconducting coil, producing a magnetic ...
0
votes
2answers
51 views

Maxwell's equations

In Jaynes' Probability Theory, he states: There are many more analogies. In physics we are accustomed to finding that any advance in knowledge leads to consequences of great practical value, but ...
2
votes
1answer
114 views

Why only gauge transformations in electromagnetism?

first of all, I need to say that I'm a mathematician, so this question may sound a little stupid. Keeping this is mind, please, try to use coordinate-free notations. Along this question, I will use ...
4
votes
2answers
781 views

Neither Biot-savart nor Ampere Law can solve this problem?

I'm confused about the use of the Ampere's Law and the Biot-Savart Law due the inconvenience of each law. I want to calculate the magnetic field due to current carrying a circular loop over itself, ...
2
votes
1answer
54 views

Homogenuous Maxwell Equations in the Language of Differential Forms

I understand that if I define electric field to be $E=E_i dx^i$, magnetic field to be $B=B_1 dx^2 \wedge dx^3 + B_2 dx^3 \wedge dx^1 + B_3 dx^1 \wedge dx^2 $, and field strength to be $F= dx^0 ...
-1
votes
2answers
64 views

Is there any property of a neutrino that prevents it from being considered the missing monopole that will make Maxwell's equations symetric

The zero in Gauss's magnetic law, is it an approximation? Could it be in reality be a really tiny number like the magnetic field strength of a neutrino? Neutrinos are members of the Lepton family ...