Explain the contradiction in Ampere�s Law

Let us consider a fundamentally different situation where a parallel-plate capacitor (A, B) is being charged as shown in Fig. If Q is the instantaneous value of charge on the capacitor, then the instantaneous value of the current in the connecting wires is I, where

\(I=frac{dQ}{dt}\) … (2)

Let us consider two surfaces S1 and S2, bounded by a closed path inside the capacitor.

The surface S2 is pierced by current I. The surface S1 is, however, not pierced by this current because this surface is in the space between the plates of the capacitor.

Clearly, for surface S2, \(oint{overrightarrow{B}}.,overrightarrow{dl}={{mu }_{0}}I,,,and\)

for surface S1, \(oint{overrightarrow{B}}.,overrightarrow{dl}=0.\)

Thus, there is an apparent contradiction in applying Ampere’s law in this case.

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