Finding the Monopole and Multipole Moments of the Electric Potential

[Athenian]

Homework Statement

The expression of the multipole moments of the electric potential as a result of a charge density function can be expressed as the below equation.

$$\rho_{\ell} ^{m} = \int r_s ^{\ell} \rho_q (\vec{r}_s) Y_\ell ^{m*} (\hat{r}_s) d^{(3)} \vec{r}_s$$

Note that $Y_\ell ^{m*}$ is the complex conjugate of the spherical harmonics $Y_\ell ^{m}$. In addition, note that $\rho_q (\vec{r}_s) = q\delta^{(3)} (\vec{r}_s)$ (i.e. the definition of a charge density at the origin for a given point charge expressed as a Dirac delta function here).

Find the monopole moment (i.e. $\rho_0^0$) for the given point charge placed at the origin.

Afterward, find the value for all the other given multipole moments here.

Relevant Equations

Refer below $\longrightarrow$

My first attempt revolved mostly around the solution method shown in this "site" or PowerPoint: http://physics.gmu.edu/~joe/PHYS685/Topic4.pdf .

However, after studying the content and writing down my answer for the monopole moment as equal to $\sqrt{\frac{1}{4 \pi}} \rho$, I found out the answer was incorrect.

To my understanding, I should plug in the value for $\rho_q (\vec{r}_s)$ (expressed as a Dirac delta function) back into the long "multipole moments" expression before doing the integral to find for $\rho_\ell ^m$.

However, I'm just stumped on how to begin solving for the integral that looks as chaotic as this (e.g. Dirac delta functions ... to the third "power").

If my line of thought is correct, after solving for the integral, I would then plug in the numerical value for $\ell$ in $\rho$ to find the values of the monopole (i.e. $\ell=0$) and multipole (i.e. $\ell > 0$) moments.

Thank you for reading through this. And, if anybody here can help me out, thank you very much!

 

[TSny]

note that $\rho_q (\vec{r}_s) = q\delta^{(3)} (\vec{r}_s)$ (i.e. the definition of a charge density at the origin
...
However, I'm just stumped on how to begin solving for the integral that looks as chaotic as this (e.g. Dirac delta functions ... to the third "power").

The "3" in the expression $\delta^{(3)} (\vec{r}_s)$ denotes the number of spatial dimensions. That is, you are working in 3-dimensional space. The 3 is not a power. (You put "power" in quotes, so maybe you weren't implying that the 3 is actually a power.)

See here for the explicit expression for $\delta^{(3)} (\vec{r}_s)$ in spherical coordinates.