Question

Explain the inconsistency of Ampere’s law during charging of a capacitor.

Answer 1

The inconsistency in Ampere’s law was that it only stood correct for some cases, that is, it was not a universally applicable law (more details in the step by step solution). Maxwell modified amperes law by including displacement current in the equation, to understand the working of charging of capacitors and thus he cleared the inconsistency of ampere's law.

Complete step by step answer:
Ampere's law:
This law states that the magnetic field produced by an electric current is proportional to the size of electric current with a constant of proportionality being the permeability of free space.
Ampere’s circuital law states that the closed line integral of a magnetic field around a current-carrying conductor is equal to absolute permeability times the total current flowing in the conductor. This can be represented mathematically as $\oint{\overset{\to }{\mathop{B.}}\,}\overset{\to }{\mathop{dl}}\,={{\mu }_{\circ }}I$ where
$\overrightarrow{B}$ represents the magnetic field, $I$ stands for the current and ${{\mu }_{\circ }}$ is the absolute permeability.
The inconsistency of this law is that it is logically incorrect.
This law is true only for steady currents. Understanding this inconsistency, Maxwell included time-varying electric fields into the law.
While charging a conductor using the current-carrying coil, there is current between the plates of the capacitor. But the ampere's law applies only when there is no current between the plates of the conductor. Thus, Maxwell modified the law.
According to Maxwell, there is the conduction of charges between the plates of the capacitor and the electric field is directed from positive plate to negative plate of the capacitor. This gives rise to displacement current. There will be current between the plates of the capacitor, called displacement current and there will be current on the surface, called conduction current.
At some points, there will be only conducting currents and no displacement current and vice versa. The pre-modified ampere’s law is valid only for the points where there is only conduction current and no displacement current.
Thus the inconsistency of Ampere’s law was cleared by Maxwell.

Note: The magnetic fields are produced both by static conduction current fields and also time-varying electric fields. The ampere law gives us the magnetic field in case of spherical, cylindrical or rectangular symmetries but this law also gave us an idea about the symmetry between electricity and magnetism.