PreprintCC BY 4.0arXiv
Voltage and current density in a rectangle with point in- and ejection via Green's functions and q-analysis
Abstract
The current density in a rectangle in the D=2 dimensional Euclidean space, where opposite point charges are placed in a source s and a destination d, is computed as the gradient of a potential that obeys the Poisson equation (first law of Maxwell). Although the computation of the current density in a 2D rectangle is a classical problem, several barriers (e.g. very slow convergence of double Fourier series) were eventually alleviated by a q-analysis, related to Gaussian polynomials and Jacobi's theta-functions, that form the basic building blocks for any elliptic function. We present an exact and computationally very efficient analytic formula of the potential and the magnitude of the current density in a rectangle.
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BibTeX
@misc{mieghem2026voltage,
title = {{Voltage and current density in a rectangle with point in- and ejection via Green's functions and $q$-analysis}},
author = {P. Van Mieghem},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26649},
eprint = {2609.26649},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26649},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Voltage and current density in a rectangle with point in- and ejection via Green's functions and q-analysis
AU - P. Van Mieghem
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26649
UR - https://arxiv.org/abs/2609.26649
AB - The current density in a rectangle in the D=2 dimensional Euclidean space, where opposite point charges are placed in a source s and a destination d, is computed as the gradient of a potential that obeys the Poisson equation (first law of Maxwell). Although the computation of the current density in a 2D rectangle is a classical problem, several barriers (e.g. very slow convergence of double Fourier series) were eventually alleviated by a q-analysis, related to Gaussian polynomials and Jacobi's theta-functions, that form the basic building blocks for any elliptic function. We present an exact and computationally very efficient analytic formula of the potential and the magnitude of the current density in a rectangle.
ER -