Question

Derivation of Maxwell's Fourth Equation!

Answer 1

Maxwell�s Fourth Equation For static electromagnetic fields, Ampere�s circuit law is nabla times H^vector = J^vector Taking divergence on both side of eqn (1) nabla . (nabla times H^vector) = nabla . J^vector But the divergence of the curl of any vector field is zero nabla . (nabla times H^vector) = 0 = nabla . J^vector nabla . J^vector = 0 The continuity equation of current requires that nabla . J^vector = (-partialdifferential f_v /partialdifferential t) notequalto 0 Thus equation (2) and (3) are obiviously in compatible for time varying conditions. We must modify equation (1) to agree with equation (3) To do this, we add a term to equation (1), so that it becomes nabla times H^vector = J^vector + J^vector _d Again taking the divergence of equation (4) nabla middot (nabla times H^vector) = nabla middot (J^vector + J^vector _d) nabla middot (nabla times H^vector) = nabla middot J^vector + nabla middot J^vector _d