Maxwell's Displacement Current in the Two Gauges

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Displacement current was originally conceived by James Clerk Maxwell in 1861 in connection with linear polarization in a dielectric solid which he believed to pervade all of space. Modern textbooks however adopt a different approach. The official teaching today is that displacement current is a consequence of extending the original solenoidal Ampère’s Circuital Law to embrace the conservation of electric charge. Yet unless either of these two methods leads to a displacement current that is compatible with Faraday’s Law of Induction, then it cannot serve its main purpose, which is to provide a bridge between Ampère’s Circuital Law and Faraday’s Law of Induction, hence enabling the derivation of the electromagnetic wave equations. This matter will be investigated in both the Lorenz gauge and the Coulomb gauge.

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