Geometrical simplification of the dipole-dipole interaction formula

TL;DR

This paper presents a simple geometric transformation of the dipole-dipole interaction formula, revealing new insights into its behavior and aiding educational understanding, including the concept of the magic angle.

Contribution

It introduces a novel geometric rewriting of the dipole-dipole potential energy formula, highlighting features like the magic angle and enhancing pedagogical approaches.

Findings

Reorientation can double the attraction between dipoles.

The transformation clarifies the origin of the magic angle.

Potential applications in educational toy design.

Abstract

Many students meet quite early this dipole-dipole potential energy when they are taught electrostatics or magnetostatics, and it is also a very popular formula, featured in the encyclopedias. We show that by a simple rewriting of the formula it becomes apparent that for example, by reorienting the two dipoles, their attraction can become exactly twice as large. The physical facts are naturally known, but the presented transformation seems to underline the geometrical features in a rather unexpected way. The consequence of the discussed features is the so called magic angle which appears in many applications. The present discussion also contributes to an easier introduction of this feature. We also discuss a possibility for designing educational toys and try to suggest why this formula has not been written down frequently before this work. Similar transformation is possible for the field of a single dipole, there it seems to be observed earlier, but also in this case we could not find any published detailed discussion.