# 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 $\overrightarrow{\bigtriangledown&space;}X&space;\overrightarrow{E}=-\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}$ , which shows that a time-varying Magnetic field produces an Electric field.

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

$\overrightarrow{\bigtriangledown&space;}&space;X&space;\overrightarrow{H}=\overrightarrow{J}$

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

$\overrightarrow{\bigtriangledown&space;}.(\overrightarrow{\bigtriangledown&space;}&space;X&space;\overrightarrow{H})=\overrightarrow{\bigtriangledown&space;}.\overrightarrow{J}$

$\overrightarrow{\bigtriangledown&space;}&space;.&space;\overrightarrow{J}=0$ ,since $\overrightarrow{\bigtriangledown&space;}.(\overrightarrow{\bigtriangledown&space;}&space;X&space;\overrightarrow{H})=0---------Equation(1)$

the divergence of the curl is identically zero which implies $\overrightarrow{\bigtriangledown&space;}.\overrightarrow{J}=0------Equation(2)$, but from the continuity equation $\overrightarrow{\bigtriangledown&space;}.\overrightarrow{J}&space;=&space;-\frac{\partial&space;\rho&space;_{v}}{\partial&space;t}-------Equation(3)$ which is not equal to zero, as $\frac{\partial&space;\rho&space;_{v}}{\partial&space;t}\neq&space;0$ is an unrealistic limitation(i.e we can not assume $\frac{\partial&space;\rho&space;_{v}}{\partial&space;t}$ as zero) .

$\therefore$ to make a compromise between the above two situations we must add an unknown term $\overrightarrow{G}$ to Ampere’s Circuital law

i.e, $\overrightarrow{\bigtriangledown&space;}X\overrightarrow{H}&space;=&space;\overrightarrow{J}+\overrightarrow{G}$

then by taking the Divergence of the above equation

$\overrightarrow{\bigtriangledown&space;}.(\overrightarrow{\bigtriangledown&space;}X\overrightarrow{H})&space;=&space;\overrightarrow{\bigtriangledown&space;}.\overrightarrow{J}+\overrightarrow{\bigtriangledown&space;}.\overrightarrow{G}------Equation(4)$

from Equation(1),Equation(4) becomes     $\overrightarrow{\bigtriangledown&space;}.\overrightarrow{J}+\overrightarrow{\bigtriangledown&space;}.\overrightarrow{G}=0$

$\overrightarrow{\bigtriangledown&space;}.\overrightarrow{G}=-\overrightarrow{\bigtriangledown&space;}.\overrightarrow{J}$

thus $\overrightarrow{\bigtriangledown&space;}.\overrightarrow{G}&space;=&space;\frac{\partial&space;\rho&space;_{v}}{\partial&space;t}---------Equation(5)$

from Maxwell’s first Equation $\overrightarrow{\bigtriangledown&space;}.\overrightarrow{D}=\rho&space;_{v}$

then Equation (5) becomes $\overrightarrow{\bigtriangledown&space;}.\overrightarrow{J}&space;=&space;\frac{\partial&space;}{\partial&space;t}&space;(\overrightarrow{\bigtriangledown&space;}.\overrightarrow{D})$

$\overrightarrow{\bigtriangledown&space;}.\overrightarrow{G}&space;=\overrightarrow{\bigtriangledown&space;}.&space;\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$

then   $\overrightarrow{G}&space;=&space;\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$

$\overrightarrow{\bigtriangledown&space;}X\overrightarrow{H}&space;=&space;\overrightarrow{J}+\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$

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 $\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$ has the dimensions of current density Amperes/Square-meter. Since it results from a time-varying electric flux density ($\overrightarrow{D}$ ) , Maxwell termed it as displacement current density $\overrightarrow{J_{D}}$.

$\overrightarrow{\bigtriangledown&space;}X\overrightarrow{H}&space;=&space;\overrightarrow{J}+\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$

$\overrightarrow{\bigtriangledown&space;}X\overrightarrow{H}&space;=&space;\overrightarrow{J}+\overrightarrow{J_{D}}$

$\overrightarrow{J_{D}}=\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$

up to this point three current densities are there $\overrightarrow{J}=\sigma&space;\overrightarrow{E}$ , $\overrightarrow{J}=\rho&space;_{v}&space;\overrightarrow{v}$ and $\overrightarrow{J_{D}}=&space;\frac{\partial\overrightarrow{D}&space;}{\partial&space;t}$.

when the medium is Non-conducting medium $\overrightarrow{\bigtriangledown&space;}X&space;\overrightarrow{H}=\frac{\partial\overrightarrow{D}&space;}{\partial&space;t}$

the total displacement current crossing any given surface is expressed by the surface integral $I_{d}&space;=&space;\oint_{s}&space;\overrightarrow{J_{D}}.\overrightarrow{ds}$

$I_{d}&space;=&space;\oint_{s}&space;\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}.\overrightarrow{ds}$

from Ampere’s law $\oint_{s}(\overrightarrow{\bigtriangledown&space;}X&space;\overrightarrow{H}).\overrightarrow{ds}=\int_{s}&space;\overrightarrow{J}.\overrightarrow{ds}&space;+\oint_{s}&space;\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}.\overrightarrow{ds}$

$\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=\int_{s}&space;\overrightarrow{J}.\overrightarrow{ds}&space;+\oint_{s}&space;\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}.\overrightarrow{ds}$

$\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=I&space;+\oint_{s}&space;\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}.\overrightarrow{ds}$

$\oint_{l}\overrightarrow{H}.\overrightarrow{dl}=I&space;+I_{d}$

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