All Questions
Tagged with electromagnetism integration
61
questions
4
votes
0
answers
119
views
The magnetic field of a spinning charged sphere
Evaluate $\displaystyle \int_{0}^{2\pi}\int_{0}^{\pi}\frac{(z_0-R\cos\theta)\sin^2\theta\cos\phi}{[(x_0-R\cos\phi\sin\theta)^2+(y_0-R\sin\phi\sin\theta)^2+(z_0-R\cos\theta)^2]^{\frac{3}{2}}}d\theta d\...
- 701
2
votes
3
answers
83
views
How to compute the volume integral for the potential of an arbitrary point outside a uniformly charged ball?
$$\frac{\rho}{4\pi\epsilon_0}\iiint_{D}^{}\frac{1}{\left\| \mathbf{r}-\mathbf{r'} \right \| }dV'$$
$D$ is a ball of radius $R$
$\mathbf{r}$ is the position vector of the point where we want to ...
- 163
2
votes
2
answers
76
views
How to apply integration by parts to simplify an integral of a cross product?
I'm reading a physics paper and am trying to figure out how a certain expression is derived (If interested, see Appendix of the paper, Eq. (A7), (A8)). The authors skip a lot of derivation steps and ...
- 23
1
vote
3
answers
152
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$\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr$
I'm trying to solve problem 3.3 from Jackson's Classical Electrodynamics, but I'm encountering some troubles solving
$$
\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr,
\qquad l = 0,2,4,6,\ldots
$$
...
- 135
6
votes
2
answers
131
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Fubini's theorem for differential forms? Why does $\int_{t_0}^{t_1}(\oint_{\partial\Omega}j)dt=\int\limits_{[t_0,t_1]\times\partial\Omega}dt\wedge j$?
In an electrodynamics book I came across the following claim for the electric current density (twisted) 2-form $j$ along the boundary of some 3-dimensional volume $\Omega$:
$$\int_{t_{0}}^{t_{1}}\left(...
- 1,490
2
votes
1
answer
85
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How to evaluate the integral $\int_{-r/2}^{r/2} \int_{-r/2}^{r/2} \frac{1}{x^2+y^2+r^2/4} dx dy$
I came across this integral while trying to evaluate the electrical force exerted by a charged plate in the form of a square with side length $r$. I tried the usual method of first keeping $y$ ...
- 409
3
votes
0
answers
47
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An improper integral from Jackson's book involving the modified Bessel function
When deriving the angular distribution of energy for synchrotron radiation one has to evaluate two tricky improper integrals (see [1] below):
$$
I_1 \equiv \int_{0}^{\infty} x^2 [K_{2/3}(x)]^2 \, \...
0
votes
1
answer
55
views
I was trying to find field due uniformly charged sheet at a distance h from centre of the square sheet
I assumed the square(side a) sheet to be made up of wires.$$dE=Kdq/r^2$$
The field due to a wire is : Reference
$$\frac{K\lambda}{d}\left[\frac{x}{\sqrt{d^2+x^2}}\right]^{(a/2)}_{(-a/2)}=\frac{K\...
- 471
0
votes
0
answers
24
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Integrals for the the localized pyramid basis functions in Galerkin Method
I tried to show the following relations for the localized pyramid basis function $\phi_{i j}(x, y)=(1-|x| /$ $h)(1-|y| / h),|x|<h,|y|<h$, where $x$ and $y$ are measured from the site $(i, j)$. ...
0
votes
0
answers
63
views
Electric field flux proportional to the field lines generated by (for example) a static charge
Suppose we have a stationary positive charge at a point in space that we call $+Q$. We know by definition that the flow of the electrostatic field is given by, in its simplified form,
$$\Phi_S(\vec E)=...
- 7,792
6
votes
1
answer
111
views
What is the value of $\frac{1}{2}\int_B\int_B\frac{\rho(x,y,z)\rho(x',y',z')}{4\pi\epsilon_0\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}dxdydzdx'dy'dz'$?
I am reading a book about electromagnetism by Yousuke Nagaoka.
Suppose $R$ is a positive real number.
Suppose $Q$ is a positive real number.
Let $B:=\{(x,y,z)\in\mathbb{R}^3:\sqrt{x^2+y^2+z^2}\leq R\}...
- 8,750
2
votes
2
answers
251
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Why can't math software solve the integral $\int\limits_{-a/2}^{a/2}\int\limits_{-a/2}^{a/2}\frac{1}{\sqrt{x^2+y^2+z^2}}dxdy$?
Consider the task of finding the electric field at a height $z$ above the center of a square sheet of side a carrying uniform charge $\sigma$.
I am asking this in the math stack exchange because ...
- 5,021
4
votes
1
answer
138
views
Finding $\int_{-\infty}^{\infty}\frac{\sin(ax)}{\sqrt{(b-x)^2+c^2}}dx$ [closed]
How do we find
$$\int_{-\infty}^{\infty}\dfrac{\sin(ax)}{\sqrt{(b-x)^2+c^2}}dx$$
I have no idea on how to even begin approaching this. Can I get a hint?
EDIT:
I've got to this integral from a physics ...
- 183
1
vote
1
answer
130
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Given a triple integral representing the (electric) vector field of a continuous volume charge distribution, how to obtain the potential function?
My question is about the math involved in the concept of electric potential. Though this is a physics concept, it is basically a lot of vector calculus. The derivations below are based on the content ...
- 5,021
1
vote
0
answers
59
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Kramers-Kronig computation for real susceptibility
i am trying to get the real part of electric susceptibility using the imaginary part with Kramers-Kronig relation for a Lorentz-Drude model.I chose to ask this question in math stack exchange as im ...
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