Exactly midway of a parallel plate capacitor a charged plate was placed having charge q. The capacitance is connected to a battery of voltage V. The plate is moved perpendicular to itself by a distance x, What is the charge flowing through the external circuit of the capacitor, if the distance between its plates equal to d?

The plate kept between a capacitor divides the capacitor into two capacitors connected in series.

For the intial configuration, the capacitance of each half of the capacitor is
Hence when connected the two capacitors are connected in series, the overall capacitance becomes.

When the plate is moved by distance, the capacitane of one side of the capacitor becomes
The capacitance of other half becomes
Hence the total capacitance becomes
Therefore, the capacitance of the system does not change even by moving the plate. So, the charge over the capacitor does not change at any time and so no current flows through the external circuit.