I know that the density and potential (in spherycals) of a charged ring is, respectively,:

$$ \rho(\textbf{r}) = \frac{\lambda}{a} \delta(r-a)\delta(\theta-\tfrac{\pi}{2}) $$

$$ \varphi(\textbf{r})= \frac{2\pi a \lambda}{r_>} \left[ 1+ \sum_{n=1}^\infty (-1)^n \frac{(2n-1)!!}{(2n)!!}\left(\frac{r_<}{r_>}\right)^{2n}P_{2n}(\cos\theta) \right] $$

Where $P_{2n}$ is the $2n$-th Legendre Polynomial, and $r_<=\min\{a,r\},r_>=\max\{a,r\}$. If I use the following formula of the electrostatic energy:

$$ U_E=\frac12\int\rho(\textbf r)\varphi(\textbf r)d^3\textbf{r} \,\propto\, \left[ 1+ \sum_{n=1}^\infty \left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \right] \to\infty $$

So this is a problem, because this is not useful. I don't know if in this case the following affirmation is right: $$ U_E=\frac12\int\rho(\textbf r)\varphi(\textbf r)d^3\textbf{r} \stackrel{?}{=} \frac{1}{8\pi} \int |E|^2 d^3\textbf{r} $$

In case of they aren't the same, can I associate this to the potential energy?

  • $\begingroup$ You have a multipole expansion of the potential, but you're assuming all of the multipole moments are equal. Is this a correct assumption? Or should you have some coefficient $c_n$ on each term? $\endgroup$ Jun 9, 2020 at 15:16
  • $\begingroup$ Why are you saying that the multipole moments are equal? I don't understand sorry. $\endgroup$ Jun 10, 2020 at 2:14
  • $\begingroup$ The expression for $\varphi$ looks like a multipole expansion. Is it not? $\endgroup$ Jun 10, 2020 at 3:23
  • $\begingroup$ @probably_someone I don't know if it's the same what i do, I wrote the Green's function like Legendre Polynomials expansion $\dfrac{1}{|\mathbf x - \mathbf x'|}= \dfrac{1}{r_>}\displaystyle\sum_{n=0}^\infty \left(\dfrac{r_<}{r_>}\right)^nP_n(\cos\gamma)$ $\endgroup$ Jun 10, 2020 at 13:29


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