A combined form of the laws of Ampere and Biot–Savart

Abstract

This article gives a combined form of the laws of Ampere and Biot–Savart which allows us to calculate the infinitesimal magnetic circulation around a closed loop ∑δB⋅δl in terms of the current flow i through an infinitesimal current path δs, and the change in the solid angle subtending the closed loop δΩ as the apex of the solid angle traverses δs:∑δB⋅δl=iδΩ×10−7. After giving its derivation from the law of Biot–Savart, the combined form is applied to a variety of magnetostatic problems. The illustrative examples, none of which requires the use of calculus for its solution, include the usual introductory course examples such as determining the magnetic field inside a toroidal coil, and at the center and along the axis of a circular current loop. In addition, the potentialities of the combined law are demonstrated by determining the field anywhere in the plane and along the axis of a rectangular current loop. Appendices include discussions of solid angles and relativistic effects.