Magnetic Flux Density of a Circular Loop Derived from the Vector Potential: A New Approach

Circular current loops are foundational to understanding electromagnetic devices like solenoids and transformers, with critical applications in metal detection, magnetic confinement, and wireless power transfer. Although the magnetic flux density field along the loop's axis is easily derived, calculating it at arbitrary off-axis points is analytically challenging, often leading to treatments that present results without a clear derivation. This study addresses this gap by providing a detailed, pedagogical calculation of the magnetic flux density, B, derived rigorously from the magnetic vector potential, A. While calculations are possible in Cartesian or spherical coordinate systems, we demonstrate that the cylindrical coordinate system is preferable, as it naturally exploits the cylindrical symmetry of the problem. This choice leads to a logically streamlined path to the solution, which is elegantly formulated in terms of complete elliptic integrals of the first and second kind. The derivation yields explicit expressions for the radial and axial field components at any point in space, while the azimuthal component vanishes due to symmetry. As a critical validation, we recover the standard elementary expression for in-plane and on-axis fields as a limiting case of the general solution. This work consolidates classical results into a unified and transparent framework that serves a dual purpose: it demystifies the underlying mathematics for students and educators. It provides a reliable, efficient analytical tool for benchmarking numerical simulations in engineering and research. By clarifying the derivation from first principles, this study strengthens the theoretical foundation for modeling circular current loops and broadens their practical applicability.