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Derivation of the Biot-Savart Law from Ampere's Law Using the Displacement Current

### Abstract

The equation describing the magnetic field due to a single, nonrelativistic charged particle moving at constant velocity is often referred to as the “Biot-Savart law for a point charge.” ^{1} Introductory calculus-based physics books usually state this law without proof. ^{2} Advanced texts often present it either without proof or as a special case of a complicated mathematical formalism. ^{3} Either way, little or no physical insight is provided to the student regarding the underlying physics. This paper presents a novel, basic, and transparent derivation of the Biot-Savart law for a point charge based only on Maxwell's displacement current term in Ampere's law. This derivation can serve many pedagogical purposes. For example, it can be used as lecture material at any academic level to obtain the Biot-Savart law for a point charge from simple principles. It can also serve as a practical example of the important fact that a changing electric flux produces a magnetic field.

© 2013 American Association of Physics Teachers

Acknowledgments:
I gratefully acknowledge the expert technical assistance of my colleague Carlos Mera in helping me prepare this manuscript for publication.

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