Scientific Basis

Abstract

Although the law by which the field created by a conductor of arbitrary shape can be calculated has been known for a long time, there are not many specific solutions. These include solutions for the axis of a circular current or the axis of a solenoid. Less frequently, a general solution for a circular current can be found. In this work, based on the Biot–Savart law, general functions (functionals) are derived, the arguments of which are parametric functions and their derivatives describing the current curve in space. The result of the operation of these functionals are the specific functions Bx, By, Bz, which describe the components of the magnetic field vector at any given point in space.

Principle of Parallel Transport of the Field Source in Space:

Let us assume that for a fixed point P(𝑥, 𝑦, 𝑧), we know the functions describing the values of the magnetic field vector components at that point: Bx(𝑥, 𝑦, 𝑧), By(𝑥, 𝑦, 𝑧), Bz(𝑥, 𝑦, 𝑧). We then parallelly move the field source along the 𝑥, 𝑦, 𝑧 coordinates by values a, b, c, respectively. Since we have reduced the distance between the point of measurement and the source by the corresponding amounts, the new functions will take the form: Bx(𝑥 − 𝑎, 𝑦 − 𝑏, 𝑧 − 𝑐), By(𝑥 − 𝑎, 𝑦 − 𝑏, 𝑧 − 𝑐), Bz(𝑥 − 𝑎, 𝑦 − 𝑏, 𝑧 − 𝑐)

Author

Ievgen Kandaurov

Higher technical education. Graduated in 1992 from KPI (Kyiv Polytechnic Institute), Engineering and Physics Faculty.