# Inconsistensy in Ampere’s law (or) Displacement Current density

Faraday’s experimental law has been used to obtain one of Maxwell’s equations in differential form  , which shows that a time-varying Magnetic field produces an Electric field.

From Ampere’s  Circuital law which is applicable to Steady Magnetic fields

By taking divergence of Ampere’s law the Ampere’s law is not consistent with time-varying fields

,since

the divergence of the curl is identically zero which implies , but from the continuity equation  which is not equal to zero, as  is an unrealistic limitation(i.e we can not assume  as zero) .

to make a compromise between the above two situations we must add an unknown term  to Ampere’s Circuital law

i.e,

then by taking the Divergence of the above equation

from Equation(1),Equation(4) becomes

thus

from Maxwell’s first Equation

then Equation (5) becomes

then

This is the equation obtained which does not disagree with the continuity equation. It is also consistent with all other results. This is a second Maxwell’s Equation is time-varying fields so the term  has the dimensions of current density Amperes/Square-meter. Since it results from a time-varying electric flux density ( ) , Maxwell termed it as displacement current density .

up to this point three current densities are there  ,  and .

when the medium is Non-conducting medium

the total displacement current crossing any given surface is expressed by the surface integral

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