On the solution arising in two-cylinders electrostatics

Area of Science:

• Physics
• Electromagnetism
• Applied Mathematics

Background:

• Understanding electrostatic interactions is crucial in various scientific and engineering fields.
• Boundary integral equation (BIE) methods offer efficient solutions for problems with complex geometries.

Purpose of the Study:

• To analyze the electrostatics of two cylinders with symmetrical or anti-symmetrical potentials.
• To explore the solution space and conditions at infinity for these electrostatic configurations.

Main Methods:

• Utilized the null-field boundary integral equation (BIE) method.
• Employed the degenerate kernel of bipolar coordinates for mathematical formulation.
• Applied the Fredholm alternative theorem to determine coefficients and analyze solution existence.

Main Results:

• Investigated the uniqueness, infinite, and no-solution cases for the electrostatic problem.
• Examined the behavior at infinity and flux equilibrium along boundaries.
• Compared findings with existing literature (Darevski, Lekner) for validation.

Conclusions:

• The BIE method effectively solves the electrostatics of charged cylinders.
• The study provides insights into the solution space and boundary conditions at infinity.
• Results are consistent with established theoretical frameworks and previous research.