Stored electrostatic energy of a uniformly charged annulus

Ciftja, Orion

doi:10.1016/j.elstat.2023.103794

Cited by 2 publications

(2 citation statements)

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“…• Uniformly charged annulus. The final result for the selfenergy is given in the following form: [75] U…”

Section: Stored Coulomb Self-energy Of Some Uniformly Charged Bodiesmentioning

confidence: 99%

“…The final expression for electrostatic self‐energy of a uniformly charged equilateral triangle is [ 74 ]

\begin{equation} E_{u} = 2 \ln (3)\,\frac{k_e \, Q^2}{L} \end{equation}

where L is the length of the equilateral triangle. Uniformly charged annulus. The final result for the self‐energy is given in the following form: [ 75 ]

\begin{eqnarray} U(\alpha) &=& \frac{8}{3\,\pi} \frac{k_e\, Q^2}{R_2}\frac{1}{(1-\alpha ^2)^2} \nonumber\\ && \times \left[1+\alpha ^3-(\alpha ^2+1) \, E(\alpha ^2) -(\alpha^2-1)\,K(\alpha^2)\right]; \nonumber\\ && \quad 0 \le \alpha \equiv R_1/R_2 &lt; 1 \end{eqnarray}

Uniformly charged cylindrical shell. Notice that the total energy of a uniformly charged cylindrical shell is known in the literature, and that it decreases rapidly with α (being divergent at

\alpha =0

).…”

Section: Other Instances Related To Equilibrium Electrostatics Onlymentioning

confidence: 99%

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Correspondence between Electrostatics and Contact Mechanics with Further Results in Equilibrium Charge Distributions

Batle,

Vlasiuk,

Ciftja

2023

Annalen der Physik

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It is common in mesoscopic systems to find instances where several charges interact among themselves. These particles are usually confined by an external potential that shapes the symmetry of the equilibrium charge configuration. In the case of classical charges moving on a plane and repelling each other via the Coulomb potential, they possess a ground state à la Thomson or Wigner crystal. As the number of particles N increases, the number of local minima grows exponentially and direct or heuristic optimization methods become prohibitively costly. Therefore the only feasible approximation to the problem is to treat the system in the continuum limit. Since the underlying framework is provided by potential theory, we shall by‐pass the corresponding mathematical formalism and list the most common cases found in the literature. Then we prove a (albeit known) mathematical correspondence that will enable us to re‐discover analytical results in electrostatics. In doing so, we shall provide different methods for finding the equilibrium surface density of charges, analytical and numerical. Additionally, new systems of confined charges in three‐dimensional surfaces will be under scrutiny. Finally, we shall highlight exact results regarding a modified power‐law Coulomb potential in the d‐dimensional ball, thus generalizing the existing literature.

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“…• Uniformly charged annulus. The final result for the selfenergy is given in the following form: [75] U…”

Section: Stored Coulomb Self-energy Of Some Uniformly Charged Bodiesmentioning

confidence: 99%

“…The final expression for electrostatic self‐energy of a uniformly charged equilateral triangle is [ 74 ]

\begin{equation} E_{u} = 2 \ln (3)\,\frac{k_e \, Q^2}{L} \end{equation}

where L is the length of the equilateral triangle. Uniformly charged annulus. The final result for the self‐energy is given in the following form: [ 75 ]

\begin{eqnarray} U(\alpha) &=& \frac{8}{3\,\pi} \frac{k_e\, Q^2}{R_2}\frac{1}{(1-\alpha ^2)^2} \nonumber\\ && \times \left[1+\alpha ^3-(\alpha ^2+1) \, E(\alpha ^2) -(\alpha^2-1)\,K(\alpha^2)\right]; \nonumber\\ && \quad 0 \le \alpha \equiv R_1/R_2 &lt; 1 \end{eqnarray}

Uniformly charged cylindrical shell. Notice that the total energy of a uniformly charged cylindrical shell is known in the literature, and that it decreases rapidly with α (being divergent at