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ZetaX
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Hello community, this is my first post and i start with a question about the famous dirac delta function.
I have some question of the use and application of the dirac delta function.
My first question is:
Using Dirac delta functions in the appropriate coordinates, express the following charge distributions as three-dimensional charge densities [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image002.gif.[/I] [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image003.gif[B][I](a)[/I][/B][I] In spherical coordinates, a charge [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image005.gifuniformly distributed over a spherical shell of radius R.So, i know the definition of dirac delta function in spherical coordinates ist [tex]\rho(\vec{r})=\sum_{k=0}^{\infty}\frac{1}{r_k^2\sin\theta}\delta(r-r_k')\delta(\theta-\theta')\delta(\phi-\phi')[/tex].
Moreover i know [tex]Q=\int\rho(\vec{r})d^3r[/tex]. So i substitute the density function into the integral and transform it into spherical coordinates.
[tex]Q=\int\int\int\sum_{k=0}^{\infty}\frac{1}{r_k^2\sin\theta}\delta(r-r_k')\delta(\theta-\theta')\delta(\phi-\phi')r^2\sin\theta drd\theta d\phi[/tex]
Now i change the sum into the integral and the limits become:
[tex]\int_0^{2\pi}\int_0^{\pi}\int_{-\infty}^{\infty}\delta(r-R)drd\theta d\phi[/tex]
So here is the problem, if i integrate i get a incorrect answer...
What is my mistake?
Greetings! :)
I have some question of the use and application of the dirac delta function.
My first question is:
Using Dirac delta functions in the appropriate coordinates, express the following charge distributions as three-dimensional charge densities [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image002.gif.[/I] [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image003.gif[B][I](a)[/I][/B][I] In spherical coordinates, a charge [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image005.gifuniformly distributed over a spherical shell of radius R.So, i know the definition of dirac delta function in spherical coordinates ist [tex]\rho(\vec{r})=\sum_{k=0}^{\infty}\frac{1}{r_k^2\sin\theta}\delta(r-r_k')\delta(\theta-\theta')\delta(\phi-\phi')[/tex].
Moreover i know [tex]Q=\int\rho(\vec{r})d^3r[/tex]. So i substitute the density function into the integral and transform it into spherical coordinates.
[tex]Q=\int\int\int\sum_{k=0}^{\infty}\frac{1}{r_k^2\sin\theta}\delta(r-r_k')\delta(\theta-\theta')\delta(\phi-\phi')r^2\sin\theta drd\theta d\phi[/tex]
Now i change the sum into the integral and the limits become:
[tex]\int_0^{2\pi}\int_0^{\pi}\int_{-\infty}^{\infty}\delta(r-R)drd\theta d\phi[/tex]
So here is the problem, if i integrate i get a incorrect answer...
What is my mistake?
Greetings! :)
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