Question

Why does a galvanometer when connected in series with a capacitor show a momentary deflection, when it is being charged or discharged?

How does this observation lead to modifying the Ampere's circuital law?

Hence write the generalised expression of Ampere's law.

Answer 1

During charging and discharging of the capacitor, there is a flow of charge from the battery towards the plates of the capacitor, which produces a conduction current in the circuit. Hence, the galvanometer present in the circuit shows momentary deflection.

As the charge on the capacitor grows, the conduction current in the wires increases. When the capacitor is fully charged, the conduction current stops flowing in the wires. During charging or discharging of the capacitor, there is no conduction current between the plates of capacitor, as they are not connected by any conduction material. Therefore, in this region, a current must exist that makes the total current in the circuit continuous. This observation leads to modifying the Ampere's circuital law. 
Thus, there must be displacement current along with the conduction current in the circuit having the property of continuity, although individually they may not be continuous.

Maxwell modified Ampere's circuital law in order to make it logically consistent. He stated Ampere's circuital law in the form,

$$\oint B^{\rightharpoonup }.{dl}^{\rightharpoonup }={\mu }_0(I+I_D)={\mu }_0(I+{\in }_0\frac{d{\varphi }_E}{dt})$$

, where

$$\frac{d{\varphi }_E}{dt}$$  is rate of change of electric flux between the plates of the capacitor.
This is the generalised expression of Ampere's law.