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Hence, as in the case of Kirchhoff's circuit laws, Ampère's circuital law would appear only to hold in situations involving constant charge density. However, the law of conservation of charge tells that The divergence of the curl of any vector field, including the magnetic field B, is always equal to zero:Ĭombining these two equations implies thatīecause is nonzero constant, it follows that Taking the divergence of both sides of Ampère's circuital law gives: Inconsistency between Ampère's circuital law and the law of conservation of charge

Which can be converted to differential form, using Stokes' theorem: In its original form, Ampère's circuital law relates the magnetic field B to the current density j: Where j is the current density (in Amperes per square meter) flowing through the surface and ρ is the charge density (in coulombs per cubic meter) at each point in the volume.įrom the divergence theorem, this relationship can be converted from integral form to differential form:Īmpère's circuital law prior to Maxwell's correction The origin of the electromagnetic wave equation Conservation of chargeĬonservation of charge requires that the time rate of change of the total charge enclosed within a volume V must equal the net current flowing into the surface S enclosing the volume: Is the refractive index of the medium, is the magnetic permeability of the medium, and is the electric permittivity of the medium. The speed of light in a linear, isotropic, and non-dispersive material medium is The magnetic constant and the vacuum permittivity are important physical constants that play a key role in electromagnetic theory. If the wave propagation is in vacuum, then

2.3 Inconsistency between Ampère's circuital law and the law of conservation of charge.2.2 Ampère's circuital law prior to Maxwell's correction.2 The origin of the electromagnetic wave equation.