All Questions
Tagged with electromagnetism grad-curl-div
13
questions
0
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0
answers
89
views
Divergence theorem with normal component of a curl to a surface
Let $\mathbf{A}$ be a vector function in $\mathbf{R}^3$ and we want to find the normal and tangent components of $\nabla \times \mathbf{A}$ on a smooth and closed surface $\Gamma$. $\mathbf{n}$ is the ...
-2
votes
1
answer
93
views
calculating curl
With
$$\tilde{\mathbf{E}}=2j\hat{y}E_me^{-jk_zz}\sin(k_xx)\quad\text{for}\quad 0<x<a$$
the phasor form of Faraday's law $\nabla\times\tilde{\mathbf{E}}=-j\omega\mu_0\tilde{\mathbf{H}}$ leads to
$...
- 29
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votes
1
answer
249
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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 ...
- 4,322
1
vote
1
answer
45
views
Calculating the divergence of electric field in standard coordinates
Given an electric field $$ \vec{E(r)} = (c/r^2 ) \hat {r} $$
I want to show that $ \nabla \cdot \vec{E} = 0$ for $ r \ne 0 $ and do the calculation in standard coordinates. For simplicity I'll ...
- 436
1
vote
1
answer
162
views
Solid angle subtended by a 3D surface from the line integral along the edge (Stokes theorem)
The solid angle subtended by the surface S at a point P is:
$$
\Omega=\iint_{S} \frac{\hat{r} \cdot \hat{n}}{r^{2}} d S
$$
where $\hat{r}$ and $\hat{n}$ are unit vectors and $r =|\vec {r}|$ is the ...
- 71
1
vote
1
answer
132
views
A laplacian working on an equation containing a laplacian and a gradient
I have an equation as follows:
$$a \Delta \mathbf{u} + \mathbf{\nabla}(\mathbf{\nabla} \cdot \mathbf{u}) = 0$$
in which $a$ is a constant, $\mathbf{u}$ is a vector, $\Delta$ is the Laplacian operator, ...
- 67
0
votes
1
answer
51
views
Is there any condition under which $\nabla\cdot F=0$ implies $F=0$?
On a physics course it was stated that
$$
\nabla\cdot\vec{D}=\rho_f=\nabla\cdot(\varepsilon_0\vec{E})
$$
and then it follows that
$$
\vec{D}=\varepsilon_0\vec{E}
$$
I know this is not generally true, ...
3
votes
4
answers
2k
views
Divergence of a radial vector field
I am reading Modern Electrodynamics by Zangwill and cannot verify equation (1.61) [page 7]:
\begin{equation}
\nabla \cdot \textbf{g}(r)=\textbf{g}^{\prime}\cdot \mathbf{\hat{r}},
\end{equation}
where ...
- 51
0
votes
1
answer
167
views
Gradient in tensor form
I found a problem which had $$\partial_i (A_i \vec{G})= (\vec{\nabla} .\vec{ A} )\vec{G}+ (\vec{A}.\nabla) \vec{G} $$ but my problem is what does $$\partial_i (A_i \vec{B})$$ even mean? it doesn't ...
- 11
1
vote
1
answer
78
views
Expression for potential vector of a central field
I know that for the central field
$$
{\bf F(x)}=\alpha\cdot\frac{\bf x}{|{\bf x}|^{3}}=\alpha\cdot\left(\frac{x_{1}}{|{\bf x}|^{3}},\frac{x_{2}}{|{\bf x}|^{3}},\frac{x_{3}}{|{\bf x}|^{3}}\right)
$$
...
- 90
2
votes
0
answers
34
views
Interface conditions on electromagnetic fields
Several authors (such as Jackson in his book "Classical Electrodynamics") state the following conditions at an interface between two different media:
$(\vec{D_2} - \vec{D_1})\cdot \vec{n} = ...
- 121
1
vote
1
answer
28
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How do we decide the sign of a an integral in the divergence theorem
Picture
From the example on the picture above, the surface of the front face of the cube lies on a positive $x$, therefore, $a_x$ is positive.
But why is $a_x$ negative in the case of the back face?
...
- 15
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votes
2
answers
78
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What is the intuition behind this divergence example
So, I am taking a course on electromagnetic theory and I would like to have a firm grasp on the basics. Now there is an example in the book that asks, Find the divergence of a position vector to an ...
- 15