Published online by Cambridge University Press: 16 October 2026
This chapter develops the near-field approximation for classical electromagnetism, in which charges interact at separations and time scales small enough that retardation and radiation can be neglected. Starting from the retarded Jefimenko integral solutions of Maxwell’s equations, it introduces an expansion parameter ωs R/c and derives near-field expressions and differential equations for the electric and magnetic fields, together with corresponding scalar and vector potentials. The chapter connects these results to electrostatics and magnetostatics, derives energy densities and Poynting’s theorem specialized to near fields, and analyzes the bound energy stored around moving point charges and localized electric and magnetic dipole sources. Finally, it applies the near-field approximation to linear electronic circuits, deriving I–V relations for ideal R, L, and C elements, formulating multielement circuit equations using Kirchhoff’s laws, and examining controlled sources and the skin effect in conducting wires, where near-field considerations reveal the frequency-dependent current distribution and effective resistance.
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