A note on Purcell's basic explanation of magnetic forces

TL;DR

This paper critically examines Purcell's relativistic explanation of magnetic forces, highlighting inconsistencies and arguing it should be viewed as illustrative rather than literal, with implications for physics education.

Contribution

The paper challenges Purcell's derivation of magnetic forces, emphasizing its limitations and advocating for clearer teaching of its illustrative nature.

Findings

Purcell's derivation has inconsistencies and paradoxes.

The approach should be considered illustrative, not literal.

Educational discussions should explicitly address these issues.

Abstract

In the 1960s, E.M. Purcell presented a basic explanation of the magnetic force experienced by a test charge moving parallel to a stationary current-carrying wire. According to Purcell's derivation, this force results from the difference between the relativistic length contraction of the distance among the stationary positive charges of the wire and the relativistic length contraction of the distance among the negative charges moving in the wire when the charges are observed in the rest frame of the test charge. The contraction difference generates a charge density imbalance that, in the rest frame of the test charge, is experienced as an electrostatic force, while, in the lab frame, is perceived as the magnetic force. In the present paper, we show that Purcell's approach is problematic since it generates inconsistencies and paradoxes. We maintain that Purcell's derivation has only an illustrative and expository value and should not be taken literally as describing something that really and physically happens in the wire. Furthermore, we believe that the difficulties pointed out here should be explicitly presented and discussed when introducing Purcell's approach in physics courses at the undergraduate and graduate levels.