Considering the case of a parallel plate capacitor being charged, show how one is required to generalize Ampere's circuital law to include the term due to displacment current.

Explanation

Ampere's law is bound to fail for nonsteady current, which can be seen with the help of charging up of capacitor `ointvecB.dvecl=mu_(0)l_(enc)` (Ampere's Law) now, we want to apply it to the amperian loop shown in diagram. How do we determine `l_(enc)` it is the total current piercing a surface that has the loop for its boundary. In this case, the wire punctures this sucface so `l_(enc)=l`, fine but what if `l` draw insttead the balloon shaped surface. NO current passes through this surface and `l` conclude that `l_(enc)=0`
`phi_(E)=EA=(q_("in"))/(epsilon_(0))`
So we can conclude there is a missing term in Ampere's circuital law, that is known as displacement current.
`i_(d)=(dQ_("in"))/(dt)=(d)/(dt)(phi_(E)epsilon_(0))=epsilon_(0)(d)/(dt)(phi_(E))`
So total current `i=i_(c)+i_(d)=i_(c)+epsilon_(0)(d)/(dt)(phi_(E))`