All Questions
Tagged with electromagnetism partial-differential-equations
39
questions
1
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1
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60
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How to prove this vector identity? [closed]
I've seen this vector identity from the book[1] in page 89,
$$ (\nabla p)\times\nu =0,\ \text{on}\ \partial\Omega,$$
where $\nu $ is the outer normal vector of $\partial \Omega$, $ p \in H_0^1(\Omega)....
- 45
0
votes
1
answer
44
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Partial differential equation with Faraday's equation
We were asked to find what equation is satisfied by $\Psi(x,y,z,t)$ given that $\textbf{B} = \nabla \times (\textbf{z} \Psi)$ and
$\textbf{E} = -\textbf{z} \frac{\partial \Psi}{\partial t}$ while ...
- 103
3
votes
1
answer
84
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Solving $2\frac{\partial S}{\partial z}+\left(\frac{\partial S}{\partial r}\right)^2=0$
I have been trying to solve this PDE $$2\frac{\partial S}{\partial z}+\left(\frac{\partial S}{\partial r}\right)^2=0$$ The solution of this equation corresponding to a spherical wave of radius of ...
- 33
1
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65
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Non-homogeneous wave equation, retarded potentials and causality
Consider the non-homogeneous wave equation in three dimensions with homogeneous initial conditions:
$$
\begin{align}
& \square f(\underline{x}, t) = g(\underline{x},t), \hspace{3mm} \underline{x} \...
- 1,179
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42
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Lorenz condition and uncoupling pde - general name for techniques of pde uncoupling
I'm reading Jackson's Electrodynamics chapter 6.2.
It is possible to reduce Maxwell's equations to
$\nabla^2 \phi + \frac{\partial}{\partial t} (\nabla \cdot A) = - \frac{\rho}{\epsilon_0}$ (6.10)
$\...
- 581
1
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0
answers
75
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Non-vanishing bondary terms of the inhomogenous wave equation
I'm trying to follow a derivation of the solution to the inhomogeneous wave equation
$$
\bigg[ \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \bigg] \psi(\vec{r},t) = - f(\vec{r},t),
$$
...
- 165
1
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59
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Existence conditions for integral transforms
Let's take the Fourier transform of Faraday's law of induction
$$ \nabla \times E = - \partial_t B $$
$$ \mathcal{F}[\nabla \times E]\ = \mathcal{F}[- \partial_t B] $$
$$ \nabla \times \mathcal{F}[E]\ ...
- 613
1
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0
answers
80
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What is Poincare's lemma applied to the coulomb gauge?
Background:
I have now studied up on differential forms.
Here are some outcomes of the study
Understand $\omega:T_p \mathbb{R^n} \rightarrow \mathbb{R}$
Can integrate m-forms
Can do derivatives of ...
- 581
7
votes
1
answer
605
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Representation of the magnetic field in 2D magnetostatics
Consider a magnetostatics problem in $\mathbb{R}^3$. The problem is governed by the following equations
$$\begin{aligned}\text{Maxwell's equations}\quad &\begin{cases}\nabla\times H(x)=J(x)\\\...
- 392
0
votes
1
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94
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How to solve a pair of coupled Poisson equations with inhomogeneous boundary conditions?
I am trying to make some code that will solve the following 2D Poisson equations:
$$\left(\frac{\partial^2}{\partial x^2}+\frac{\partial}{\partial y^2}\right)P(x,y) = f(x,y),$$
$$\left(\frac{\partial^...
- 1,027
3
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0
answers
81
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Solving cylindrically symmetric non-homogeneous wave equation $\nabla^2\mathbf{A}(s,t)-\frac{\partial^2}{\partial t^2}\mathbf A(s,t)=\mathbf J(s,t)$
I'm trying to solve a non-homogeneous wave equation in cylindrical coordinates
\begin{align}
\nabla^2\mathrm A-\frac{\partial^2\mathrm A}{\partial t^2}=\mathrm J,
\end{align}
where A and J are ...
- 1,685
1
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0
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59
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References for magnetic Sobolev spaces
I am working on PDE. Recently I have been studying magnetic Sobolev spaces. While the theory is clear to me, having very little knowledge about physics, I have almost no idea how these spaces help ...
- 31
1
vote
1
answer
153
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Variable separation method for solving wave equation
In variable separable method we assume the solution to be the product of such functions each of which is function of only one variable. What is the basis for that assumption? What allows us to assume ...
3
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47
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Gauge condition in Maxwell's equation
Maxwell's equations (differential forms formulation) read
$$
dF = 0 \\
\partial^a F_{ab} = -j_b
$$
where $j_b$ is the current-density 1-form. The first equation tells us there is some 1-form $A$ so ...
- 5,141
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179
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Rigorous formulation of electromagnetic field theory for a system of moving charges (non relativistic) using distribution theory
Suppose we have a system of $n$ charged particles with trajectories given by a paths $x_j:\mathbb{R}\to\mathbb{R}^3$ then the Maxwell equations for this system are given first by defining the charge ...
- 144