All Questions
16
questions
0
votes
1
answer
75
views
What integral is used to calculate the electric field generated by a continuous charged curve?
I'm studying Multivariable Mathematics, by Ted Shifrin, in which one reads that ''the gravitational force exerted by a continuous mass distribution $\Omega$ with density function $\delta$ is
$$\mathbf{...
0
votes
0
answers
34
views
Boundary Conditions on the Magnetic Flux Density (B-field)
My question is similar to this one (Boundary conditions magnetic field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically ...
- 665
1
vote
2
answers
77
views
What does $\vec{\nabla}^2 \vec{E} = \vec{\nabla}^2 \left[ f(\vec{k} \cdot \vec{r} - \omega t) \vec{E}_0 \right]$ mean?
$\vec{E} = f(\vec{k} \cdot \vec{r} - \omega t) \vec{E}_0$ with the constant vector field $\vec{E}_0$
I only know the case if I apply the Laplacian operator on a scalar field, in this case it is a ...
0
votes
0
answers
63
views
Vector Line Integral For Biot Savart Law
How would one go about computing the vector line integral presented in the Biot-Savart law: $$\vec{B}=\int_c\frac{\mu_0I}{4\pi} \frac{d\vec{l}\times\hat{r}}{r^2}$$
I know how to compute vector line ...
- 33
8
votes
1
answer
711
views
A calculus problem from electrostatics
Since this problem consists of multiple parts and one needs to see all of them to understand the problem i'm going to list out all of them:
Consider a uniformly charged spherical shell of radius $R$ ...
- 429
0
votes
0
answers
34
views
Translational invariance of sources/materials implies translational field invariance
Let's say we have an electromagnetic problem where $\epsilon = \epsilon(x,y)$, $\rho = \rho(x,y)$, and $J = J(x,y)$. The physicist argument is that by the symmetry of the problem we also have $E = E(x,...
- 613
3
votes
2
answers
318
views
Is curl of a particle's velocity zero?
The question
Consider the motion of a particle specified by $\mathbf{x} (t): \mathbb{R} \mapsto \mathbb{R}^3$, where $\mathbf{x} = (x_1,x_2,x_3)$ in cartesian coordinates. The curl of its velocity $\...
- 33
0
votes
1
answer
249
views
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
0
votes
0
answers
76
views
Electric field with $\nabla \times \vec{E} = 0$ and $\nabla \cdot \vec{E} = 0$ outside of a conductor circulates a constant electric current
Q: suppose that I know that outside of some conductor circulates a constant electric current , I have $\nabla \times \vec{E} = 0$ and $\nabla \cdot \vec{E} = 0$. How do I prove that $\vec{E} = 0 $ ...
- 95
0
votes
1
answer
72
views
Bound Surface current across boundary of Amperian loop
For a Bound surface current $\vec{K}_{b}$ 'flowing' across through the boundary of an Amperian loop is reflected by the expression $\int \vec{K}_{b}dl_{\perp}$
What does $\int \vec{K}_{b}dl_{\perp}$ ...
- 6,343
0
votes
0
answers
1k
views
Bound surface current density on rotating sphere.
For solid sphere of radius R, azimuthal angle $\phi$ and polar angle $\theta$ rotating at velocity $\vec{v}$ with uniform surface charge $\sigma$ ,
the bound surface current density is
$\vec{K}=\...
- 6,343
3
votes
2
answers
941
views
What is an unit area vector?
As the title suggests,
What is a unit area vector?
I've tried googling but unable to arrive at any satisfactory answers.
Any help is appreciated.
- 6,343
0
votes
0
answers
40
views
Validating Stokes theorem for Faradays law.
Faradays Law can be written as
$\int_{C} E\dot{}dl = -\frac{1}{c}\frac{d}{dt}\int_{S} d^2r' n \dot{}B$ ,where left integral is a contour integral, and right is a surface integral.
Using two surfaces ...
- 663
1
vote
1
answer
48
views
Divergence theorem on vector function generated by integral
Suppose we define the magnetic field as $B(r) = \frac{1}{c}\int_{V}d^3r' \frac{J(r') \times(r-r')}{|r-r'|^3} $
Show that $\nabla \dot{}B=0$
I tried applying divergence theorem.
$\int_{V} \nabla \...
- 663
0
votes
0
answers
56
views
Difficulties employing a line integral of a half circle.
I need to use the following equation (Bio-Savart) of a wire carrying a current, where $d\vec l$ is an infinitesimal piece of wire carrying the charge (the thicker line, half circle)
:
$$d\vec B=\...
- 1,434