Exploring Dirac Delta Function: Using to Express 3D Charge Distributions

[ZetaX]

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 $$\rho(\vec{r})=\sum_{k=0}^{\infty}\frac{1}{r_k^2\sin\theta}\delta(r-r_k')\delta(\theta-\theta')\delta(\phi-\phi')$$.

Moreover i know $$Q=\int\rho(\vec{r})d^3r$$. So i substitute the density function into the integral and transform it into spherical coordinates.

$$Q=\int\int\int\sum_{k=0}^{\infty}\frac{1}{r_k^2\sin\theta}\delta(r-r_k')\delta(\theta-\theta')\delta(\phi-\phi')r^2\sin\theta drd\theta d\phi$$
Now i change the sum into the integral and the limits become:

$$\int_0^{2\pi}\int_0^{\pi}\int_{-\infty}^{\infty}\delta(r-R)drd\theta d\phi$$
So here is the problem, if i integrate i get a incorrect answer...

What is my mistake?

Greetings! :)

 

[ecastro]

Shouldn't be your limits of integration for $r$ would be from $0$ to $R$?

 

[Simon Bridge]

i.e. does it make sense for the radius to have a negative value?
Make sure you understand what each term in the notation is for.

What does the integral need to look like to come out right?
You certainly don't have the right units - you seem to have used $\rho(\vec r)$ in the first two equations to mean different things. In the first, it is the dirac delta function in spherical coordinates and in the second it is a charge density ... then you treat the dirac delta function as a charge density which it is not.