When studying (or teaching) classical electromagnetism, one is bound to deal with the electric field of an ideal electric dipole, as well as its magnetic counterpart. A careful analysis then reveals that each of those fields must include, for consistency, a term proportional to a Dirac delta function localized at the position of the dipole. However, one is usually told not to worry about those terms since, as classical interactions always involve sources which are spatially separated, the Dirac-delta terms are only relevant for quantum mechanics, where they are directly related to important phenomena. In this work, we pose and solve a purely classical problem in electrostatics in which the Dirac-delta terms in the dipole fields are indispensable. It involves the computation of the interaction energy between a conductor with a spherical cavity and an (ideal) electric dipole located at the center of that cavity. We also solve its magnetic counterpart, replacing the conductor with a superconductor and the electric dipole with a magnetic one.

中文翻译：

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在经典相互作用中，偶极场中的狄拉克三角洲可以被忽略吗？

在研究（或教授）经典电磁学时，必然要处理理想电偶极子的电场及其对应的磁偶极子。然后仔细分析表明，为了一致性，每个场都必须包含与偶极子位置处的狄拉克δ函数成比例的项。然而，人们通常被告知不要担心这些项，因为经典相互作用总是涉及空间上分离的源，狄拉克-δ项仅与量子力学相关，它们与重要现象直接相关。在这项工作中，我们提出并解决了静电学中的一个纯粹的经典问题，其中偶极场中的狄拉克-δ项是不可或缺的。它涉及计算具有球形空腔的导体与位于该空腔中心的（理想）电偶极子之间的相互作用能。我们还解决了它的磁对应物，用超导体代替导体，用磁偶极子代替电偶极子。