quaternion-letter

A paper, in the form of a letter, to examine Ampere in Quaternion form.

Sequence

Annus mirabilis - Paper #1.

Title

Ampere's Parallel Circuital Currents in Quaternion Notation

Abstract

Physics is content with understanding of concepts introduced by its earliest practitioners, especially when corresponding mathematics have demonstrated its value. One of such value is the case of action of electric currents, first detailed by Ampere in Theorie des Phenomenes Electrodynamiques. While this paper does not pose to challenge this work, it is instead curious as to its formalism; if additional value in both explicit and intuitive capacities can be offered by writing in Quaternion notation.

Introduction

Ampere's experimental law is stated as follows:

1. Equal and opposite currents in the same conductor produce equal and opposite effects upon other conductors.
2. The effect of a conductor bent or twisted in any manner is equivalent to that of a straight one, provided that the two are traversed by equal currents and the former nearly coincides with the latter.
3. No closed circuit can set into motion an element of a circular conductor about an axis through the center of the circle and perpendicular to its plane.
4. In similar systems traversed by equal currents the forces between them are equal.
5. The action between two elements of currents is a straight line intersecting them.
6. It is taken for granted that the effect of any element of a current upon another is directly the product of the strengths of the currents and of the lengths of the elements.

Mathematical Proposition

The case is stated thusly: Let there be two closed currents whose strengths are $a$ and $a_1$ let $\alpha\prime$, $\alpha_1$ be elements of these, $\alpha$ being the vector connecting their midpoints. Then the effect of $\alpha\prime$ on $\alpha_1$ must, when resolved along $\alpha_1$, be a complete differential with respect to $\alpha$ comprised as three independent variables as the total resolved effect of the closed circuit of which $\alpha\prime$ is an element is zero by (3).

The formalism surrounding Quaternions is best espoused by Tait (1890). In elaborating the case, the introduction of tensor $T$, scalar $S$ and unit-tensor $U$ is:

• tensor
• scalar
• unit tensor

New year is 29.01.2025 - 13:35:52

Publisher

• Physics Letters A
• Article Submission