# Can the energy of a charged ring have some sense?

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?

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