If a dielectric slab between the parallel plates of a capacitor is replaced by a metal plate of same thickness t < d where d is the separation between the plates of the capacitor, how does its capacitance change?

Dear Student,

Capacitance of the capacitor initially, when there is no metal plate is,

$C=\frac{{\epsilon }_{0}A}{d}$
Now, if a metal plate of thickness t is introduced between the plates of the capacitor. Then the distance between the plates of a capacitor becomes (d-t)​ because, the dielectric constant for conductor is infinity. So, the capacitance becomes,

$C\text{'}=\frac{{\epsilon }_{0}A}{d-t}⇒C\text{'}=\left(\frac{d}{d-t}\right)\left(\frac{{\epsilon }_{0}A}{d}\right)\phantom{\rule{0ex}{0ex}}⇒C\text{'}=\left(\frac{d}{d-t}\right)C$