Question

Explain inconsistency of Ampere's circuital law during charging of a capacitor.

Answer 1

Inconsistency of ampere's law Maxwell explain the ampere's law is valid only for steady current or when the electric field does not change the time. To see this inconsistency consider a parallel plate capacitor being charged by a battery.During the charging time varying current flows through connecting wires applying to amperes law for loop $l_{1}and{l}_2$

$\underset{l_1}{\oint }\overrightarrow{B.}\overrightarrow{dl}={\mu }_{0}i$

$But,\underset{l_2}{\oint }\overrightarrow{B.}\overrightarrow{dl}=0$

(since no current flows through the region between the plates). But practical it is observed that there is a magnetic field between the plates.Hence ampere's law fails i.e. $\underset{l_1}{\oint }\overrightarrow{B.}\overrightarrow{dl}\ne {\mu }_{0}i$ .

Modified ampere's circuital law vampires maxwell's circuital law: Maxwell assume that some sort of current must be flowing between the capacitor plates during charging process.He named it the displacement current. hence modified law is as follows:

$\underset{l_1}{\oint }\overrightarrow{B.}\overrightarrow{dl}={\mu }_o\left(i_c+i_d\right)$ or $\underset{l_1}{\oint }\overrightarrow{B.}\overrightarrow{dl}={\mu }_0\left(i_c+{\epsilon }_0\frac{d{\phi }_E}{dt}\right)$

where $i_c$=to conduction current=current due to flow of charges in a conductor and

$i_d$ = displacement current= ${\epsilon }_o\frac{d{\phi }_E}{dt}$ current due to the changing electric field between the plates of the capacitor