Question

Write the mathematical equation of ‘Ampere-Maxwell law’.

Answer 1

Hint:Idea of displacement current. According to him even time- varying electric fields in free space can produce a magnetic field and hence made modification in Ampere's law.

Complete step by step answer:
 Maxwell predicted that not only a current flowing in a conductor produces a magnetic field but also a time varying electric field (i.e. changing electric field) in a vacuum/free space (or in a dielectric) produces a magnetic field. It means a changing electric field gives rise to a current which flows through a region so long as the electric field is changing there. Maxwell also predicted that this current produces the same magnetic field as a conducting current can produce. This current is known as "displacement current".
Thus displacement current is that current which comes into play in the region in which the electric field and hence the electric flux is changing with time as
$I_D={\in }_o{\displaystyle \frac{d{\varphi }_E}{dt}}$
Where ${\varphi }_E$ is the electric flux.
Maxwell also found that conduction current (I) and displacement current $I_D$together have the property of continuity, although individually, they may not be continuous.
This idea led Maxell to modify Ampere circuital law in order to make the same logically consistent. He stated Ampere circuital law to the
From, $\underset{C}{\oint }\overrightarrow{B}.d\overrightarrow{l}={\mu }_o(I+I_D)$
$={\mu }_o\left(I+{\in }_o{\displaystyle \frac{d{\varphi }_E}{dt}}\right)$
It is now called the Ampere-Maxwell law.

Note:Ampere-Maxwell law explains continuity of flow of current by help of explaining displacement current. Through Ampere-Maxwell law, Maxwell made this law consistent and here he made use of changing flux in case of displacement current.