This chapter presents Ampère’s law, a cornerstone of magnetostatics that provides a powerful method for calculating magnetic fields, especially for current distributions with high symmetry. It presents both the differential and integral forms of Ampère’s law, demonstrating how the circulation of the magnetic field around a closed loop is directly proportional to the enclosed current. The chapter emphasizes the application of symmetry arguments (invariance under translation, rotation, and mirror symmetry/antisymmetry) to simplify magnetic field calculations. A significant portion is dedicated to the magnetic vector potential ( \(\textbf{A}\) ), defined such that \(\textbf{B}=\mathbf {\nabla }\times \textbf{A}\) . The chapter shows how \(\textbf{A}\) can be calculated from current distributions and discusses its non-uniqueness, leading to the concept of gauge transformations and the choice of the Coulomb gauge ( \(\mathbf {\nabla }\cdot \textbf{A}=0\) ), under which \(\textbf{A}\) satisfies a Poisson-like equation. The profound physical significance of the vector potential, even in regions where the magnetic field is zero, is highlighted through the Aharonov–Bohm effect, an experimental confirmation of quantum mechanical principles. The chapter then formalizes the two fundamental laws of magnetostatics: Gauss’s law for magnetism ( \(\mathbf {\nabla }\cdot \textbf{B}=0\) ), which signifies the absence of magnetic monopoles, and Ampère’s law. These two laws, together with the Helmholtz decomposition theorem, are shown to uniquely determine the magnetic field. Finally, the chapter introduces the magnetic dipole, defining its magnetic moment ( \(\textbf{m}\) ) for both current loops and arbitrary current distributions. It demonstrates that the magnetic field far from a localized current distribution resembles that of a magnetic dipole and derives the expressions for the forces and torques experienced by a magnetic dipole in an external magnetic field.

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Ampère’s Law and Magnetic Dipole

  • Fabian Cadiz,
  • Arnaud Couairon

摘要

This chapter presents Ampère’s law, a cornerstone of magnetostatics that provides a powerful method for calculating magnetic fields, especially for current distributions with high symmetry. It presents both the differential and integral forms of Ampère’s law, demonstrating how the circulation of the magnetic field around a closed loop is directly proportional to the enclosed current. The chapter emphasizes the application of symmetry arguments (invariance under translation, rotation, and mirror symmetry/antisymmetry) to simplify magnetic field calculations. A significant portion is dedicated to the magnetic vector potential ( \(\textbf{A}\) ), defined such that \(\textbf{B}=\mathbf {\nabla }\times \textbf{A}\) . The chapter shows how \(\textbf{A}\) can be calculated from current distributions and discusses its non-uniqueness, leading to the concept of gauge transformations and the choice of the Coulomb gauge ( \(\mathbf {\nabla }\cdot \textbf{A}=0\) ), under which \(\textbf{A}\) satisfies a Poisson-like equation. The profound physical significance of the vector potential, even in regions where the magnetic field is zero, is highlighted through the Aharonov–Bohm effect, an experimental confirmation of quantum mechanical principles. The chapter then formalizes the two fundamental laws of magnetostatics: Gauss’s law for magnetism ( \(\mathbf {\nabla }\cdot \textbf{B}=0\) ), which signifies the absence of magnetic monopoles, and Ampère’s law. These two laws, together with the Helmholtz decomposition theorem, are shown to uniquely determine the magnetic field. Finally, the chapter introduces the magnetic dipole, defining its magnetic moment ( \(\textbf{m}\) ) for both current loops and arbitrary current distributions. It demonstrates that the magnetic field far from a localized current distribution resembles that of a magnetic dipole and derives the expressions for the forces and torques experienced by a magnetic dipole in an external magnetic field.