Electric potentials and field lines for uniformly-charged tube and cylinder expressed by Appell's hypergeometric function and integration of $Z(u|m) \mathrm{sc}(u|m)$

TL;DR

The paper derives closed-form expressions for electric potentials and field lines in axisymmetric geometries (uniformly charged cylinder and tube) using elliptic integrals and Appell's hypergeometric function $F_2^{\text{Appell}}$. A novel field line potential $\psi$ is introduced, whose contour lines coincide with electric field lines outside the charged region; in the tube case $\psi$ is multivalued and carries a topological charge, with jumps $\Delta\psi=2Q$ where $Q$ is the total charge. A key by-product is an Appell-function representation for the integral $\int_0^u Z(u|m)\operatorname{sc}(u|m)\,du$, filling a gap in classical elliptic-function tables. An Addendum discusses references and a proposed decomposition by degrees of transcendence, highlighting a structured view of solutions in terms of elementary, elliptic, and Appell/higher-transcendence components. Overall, the work connects classical electrostatics to higher transcendental functions, providing exact, general tools for analyzing axisymmetric charge distributions and their field-line topologies.

Abstract

The closed-form expressions of electric potentials and field lines for a uniformly-charged tube and cylinder are presented using elliptic integrals and Appell's hypergeometric functions, where field lines are depicted by introducing the concept of the field line potential in axisymmetric systems, whose contour lines represent electric field lines outside the charged region, thought of as an analog of the conjugate harmonic function in the presence of non-uniform metric. The field line potential for the tube shows a multi-valued behavior and enables us to define a topological charge. The integral of $Z(u|m)\operatorname{sc}(u|m)$, where $ Z $ and $ \operatorname{sc} $ are the Jacobi zeta and elliptic functions, is also expressed by Appell's hypergeometric function as a by-product, which was missing in classical tables of formulas. In the Addendum appended after the main article, several relevant references are provided and the decomposition of the solution by ``degrees of transcendence'' is proposed. The attached supplementary calculations provide detailed derivations of several formulas including the integral in the title and discuss the resemblance between the field line potential for tube and the electric potential for disk.

Figures (3)

• Figure 1: The electric potential and field line potential made by a uniformly charged cylinder. Figures (a) and (b) represent $\phi^{\text{cyl}}$ and $\psi^{\text{cyl}}$ with parameters $\rho_0=1,\ R=1,$ and $Z=0.7$. The potential $\phi^{\text{cyl}}$ is $C^1$ on the surface. Figure (c) is a contour plot, showing that equipotential lines of $\phi$ and $\psi$ are orthogonal. Inside the cylinder, $\psi$ is not given by Eq. (\ref{['eq:psidef']}), so we leave it unplotted, though the field lines themselves do exist.
• Figure 2: The electric potential and field line potential made by a uniformly charged tube. We set parameters $\sigma_0=1,\ R=1,$ and $Z=0.7$. Figure (a) represent $\phi^{\text{tube}}$, which has a non-differentiable corner at $r=R,\ |z|<Z$. In figure (b), we plot $\psi^{\text{tube}}+n\Delta\psi^{\text{tube}}$ with $n=0,\pm1$, showing the log-like multi-valued character. Figure (c) is a contour plot.
• Figure 3: Numerical check of Eq. (\ref{['eq:intZsc']}) using AppellF2 in Mathematica 14.2. Here we set $m=0.85 \ [K(m)\simeq 2.39]$ and the plot range is $-4K(m)\le u \le 4K(m)$. The jump value at $u=(2n+1)K(m), n\in\mathbb{Z}$, is given by $\Delta=\frac{\pi}{2K(m)\sqrt{1-m}}\simeq 5.33.$