From the Biot-Savart Law to Ampere's Magnetic Circuital Law via Synthetic Differential Geometry

Abstract

It is well known in classical electrodynamics that the magnetic field given by a current loop and the electric field caused by the corresponding dipoles in sheets are very similar, as far as we are far away from the loop, which enables us to deduce Ampere's magnetic circuital law from the Biot-Savart law easily. The principal objective in this paper is to show that synthetic differential geometry, in which nilpotent infinitesimals are in abundance, furnishes out a natural framework for the exquisite formulation and its demonstration. This similitude in heaven enables us to transit from the Biot-Savart law to Ampere's magnetic circuital law like a shot on earth.