All Questions
92
questions
13
votes
2
answers
5k
views
Deriving Biot-Savart Law from Maxwell's Equations
As an exercise, I've been trying to derive the Biot-Savart law from the second set of Maxwell's equations for steady-state current
$$\begin{align}&\nabla\cdot\mathbf{B}=0&&\nabla\times\...
- 838
2
votes
1
answer
957
views
Maxwell equations from Euler-Lagrange equation: I keep obtaining the wrong equation
I'm deriving the Maxwell equations from this Lagrangian:
$$ \mathscr{L} \, = \, -\frac{1}{4} F^{\mu \nu}F_{\mu \nu} + J^\mu A_\nu
\tag{1}$$
My signature is $$(+ - - -)\tag{2}$$ and
$$
F^{\mu \nu} \,...
- 2,347
0
votes
1
answer
384
views
How to predict the direction of electric field generated by a changing magnetic field (in conditions of cilindrycal sysmmetry) and viceversa?
Consider the two Mawell equations
$$∇×E=-∂B/∂t \tag{1}$$
$$∇×B=μ_0 ε_0 ∂E/∂t \,\,\, \mathrm{if} \,\, i=0 \tag{2}$$
Consider the following situation with cilindrycal symmetry. A magnetic field $B$ ...
- 2,617
1
vote
1
answer
1k
views
Variation of electromagnetic action to obtain Maxwell's equations
The electromagnetic action is given in the language of differential forms by
$$S[A]=-\frac{1}{4}\int F\wedge \star F$$
The variation of the electromagnetic action $S$ gives us Maxwell's equations
$$d\...
- 3,203
1
vote
1
answer
642
views
Calculating Lagrangian of electromagnetism [duplicate]
I know that the interaction terms of the Lagrangian of electromagnetism are given by
$$L_{int}=-q\phi (\mathbf{x},t)+q\mathbf{v}(t)\cdot \mathbf{A}(\mathbf{x},t).$$
The above equation is replaced by ...
- 223
1
vote
1
answer
250
views
Question about tensor form of Maxwell equation [closed]
By variating the Maxwell Lagrangian we get the equation of motion. The remaining two Maxwell equations can be written as
$$\epsilon_{\mu\nu\rho\sigma}\partial^{\rho} F^{\mu\nu} = 0.$$
I have also seen ...
- 233
3
votes
0
answers
206
views
Confusion in reaction force of Ampere's Force Law [closed]
I am reading Maxwell's "A Treatise on Electricity and Magnetism" and I have some confusion in the following pages:
The element ds is resolved into its components $\alpha$ and $\beta$;and the element ...
- 772
1
vote
1
answer
2k
views
Rewriting Maxwell's equation in tensor form [closed]
Suppose $F_{ij}=\epsilon_{ijk}B_k $, how to prove the following:
$\partial_iB_i=0$ becomes $\partial_iF_{jk}+\partial_jF_{ki}+\partial_kF_{ij}=0$
$B_iB_i$ becomes $F_{ij}F_{ij}/2$
I can see that it ...
- 233
5
votes
1
answer
570
views
Confusion in Maxwell's derivation of Ampere's Force Law - Part II [closed]
I am reading Maxwell's "a treatise on electricity and magnetism, Volume 2, page 156" about "Ampere's Force Law". I have some confusion in the following pages:
My question is of two parts:
1. ...
- 772
0
votes
1
answer
247
views
Derivatives involving four vectors [closed]
The Schrödinger lagrangian for complex fields is
$$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}$$
Where $D_\...
1
vote
0
answers
106
views
Time dependance of oscillating sheet of charge
I am working on a practice problem involving maxwells equations. We have an infinite sheet of charge density $\sigma$ in the x-y plane and it is oscillating as x= $\Re [x_0e^{-i\omega t}]$. I want to ...
- 1,788
3
votes
1
answer
527
views
How is Biot-Savart law verification of Maxwell's 4th equation for steady current?
Please provide some theoretical procedure which equates Biot-Savart law with the Maxwell's 4th equation for steady current, which is Ampere's law
$$\quad \nabla\times{\bf B} = \mu_0{\bf J}.$$
- 2,326
0
votes
2
answers
312
views
Maxwell equations in 2+1 D
I have a problem with the Maxwell equations in (2+1) dimensions using differential form. Following J. Baez "Gauge Fields, Knots and Gravity" page 93 (or any other book), the equations are
\begin{...
- 305
5
votes
1
answer
122
views
Why do these calculations of EM fields for a magnet and wire loop seem inconsistent?
Suppose you have a conducting circular wire loop and a magnet moving towards each other. They move along the $z$ direction with nonrelativistic constant speed $v$. Let the $B$ field of the magnet in ...
- 145
0
votes
4
answers
427
views
How to propagate $E_x (x,t) = \exp(-t^2/\tau^2-i\omega_0 t) \exp(-x^2/w_0^2)$ in finite difference time domain (FDTD) analysis
Finite difference time domain (FDTD) allows to solve differential equations for time evolution.
For example, we can analyze ultra-short pulses in free space by solving the Maxwell's equations.
The ...
- 1,047