# Maxwell's Displacement Current

Maxwell's Displacement Current

According to Ampere's law, the line integral of the magnetic field around a closed loop equals the total free current passing through the loop: . In the Demonstration graphic, magnetic lines of force are shown as blue loops while the free electric current is shown as a red arrow. Ampere's law can be expressed in differential form as , where is the current density. The vector identity implies the steady-state limit of the equation of continuity: . Maxwell recognized that the more general form of the equation of continuity , where is the charge density, requires a modification of Ampere's law. Substituting from the first of Maxwell's equations, , Ampere's law can be generalized to , which becomes the third of Maxwell's equations. The added term is known as the displacement current since it involves the rate of change of the dielectric displacement . This provides a mechanism whereby a time-varying electric field can create a magnetic field, complementary to Faraday's law, in which a time-varying magnetic field can produce an electric field. What Maxwell called the "mutual embrace" of electric and magnetic fields can produce propagating electromagnetic waves. This would not be possible without the displacement current.

H

∮H·ds=

I

free

∇H=

J

free

J

∇·∇H=0

∇·J=0

∇·J+=0

∂ρ

∂t

ρ

∇·D=ρ

∇H=J+

∂D

∂t

D