[Paper Review] Circular plate capacitor with different disks

Giampiero Paffuti, Enrico Cataldo+2 more|arXiv (Cornell University)|Jul 19, 2016

Synthesis and properties of polymers12 references6 citations

TL;DR

This paper generalizes Love's integral equation for circular plate capacitors to the case of two coaxial disks with different radii, deriving the complete asymptotic form of the capacitance matrix for both small and large separations. The key contribution is a rigorous generalization of Kirchhoff's formula for the small-distance limit, validated by asymptotic analysis and integral equation solutions, providing exact expressions for capacitance coefficients including logarithmic and inverse-distance divergences.

ABSTRACT

In this paper we write a system of integral equations for a capacitor composed by two disks of different radii, generalizing Love's equation for equal disks. We compute the complete asymptotic form of the capacitance matrix both for large and small distances obtaining a generalization of Kirchhoff's formula for the latter case.

Motivation & Objective

• To extend Love's integral equation framework from equal to unequal circular disks in a coaxial configuration.
• To derive the complete asymptotic behavior of the capacitance matrix for small and large separations between disks of different radii.
• To generalize Kirchhoff's classical formula for capacitance in the small-distance limit to the case of unequal disks.
• To analyze the divergence structure of capacitance coefficients as the separation vanishes, particularly distinguishing bulk and fringe contributions.
• To provide a formal proof of the integral equation system and its asymptotic solution for electrostatic potential and charge distribution.

Proposed method

• Formulates a system of integral equations for the electrostatic potential on two coaxial disks of different radii, generalizing Love's equation for equal disks.
• Uses a perturbative approach in the small-distance limit, introducing a small parameter κ = d/(2a) where d is the separation and a is the smaller radius.
• Derives the leading-order asymptotic solution for the surface charge density via a decomposition into a leading singular part and a correction term.
• Applies a Green's function method and singular integral equation techniques to solve for the correction terms, ensuring consistency with physical constraints on charge continuity.
• Performs a controlled asymptotic expansion in the limit κ → 0, extracting logarithmic and inverse-distance divergences.
• Validates the results by showing that the limit b → 1 (where b = a₂/a₁) reproduces the known logarithmic divergence for equal disks, confirming consistency with established theory.

Experimental results

Research questions

• RQ1 How does the capacitance matrix of two coaxial circular disks with different radii behave in the small separation limit?
• RQ2 Can Love’s integral equation formulation be generalized to unequal disks, and what form do the resulting equations take?
• RQ3 What is the nature of the divergences in the capacitance coefficients as the separation between disks vanishes?
• RQ4 How do the bulk (1/ℓ) and fringe (log ℓ) contributions to capacitance scale in the unequal-disk case?
• RQ5 Does the asymptotic solution reproduce the known Kirchhoff formula for equal disks in the limit of equal radii?

Key findings

• The capacitance matrix is fully determined in the small-distance limit via a generalized integral equation system, extending Love’s result to unequal disks.
• The leading-order divergence in the self-capacitance C₁₁ is proportional to 1/κ = a/d, confirming the expected 1/d bulk scaling for the smaller disk.
• A logarithmic divergence log(1/κ) appears in C₁₁, matching the known behavior for equal disks when the radius ratio b → 1.
• The mutual capacitance C₁₂ is found to be C₁₂ = −C₁₁ + (a/π)(b − √(b²−1)), showing a non-trivial dependence on the radius ratio b.
• The solution for the surface charge density exhibits a singular behavior near the edge of the smaller disk, consistent with the 1/√(1−t²) form in the limit of small separation.
• The interchange of limits κ→0 and b→1 reproduces the correct logarithmic singularity for equal disks, confirming consistency and correctness of the asymptotic expansion.