A parallel plate air filled capacitor of capacitance \(C\) has plate area \(A\) and the distance between the plates \(d\). When a metal sheet of thickness \(\left(\frac{d}{2}\right)\) and of the same area \(A\) is introduced between the plates, its capacitance becomes \(C_2\). The ratio \(C_2: C_1\) is

The capacitance of the air filled parallel plate capacitor is given by

When a slab of dielectric constant \(K\), and thickness \(\mathrm{t}\) is introduced in between the plates of the capacitor, its new capacitance is given by \(C^{\prime}\). This leads to parallel combination of two capacitors:
\(\frac{1}{C^{\prime}}=\frac{d-t}{\varepsilon_0 A}+\frac{t}{K \varepsilon_0 A}=\frac{d+t\left(\frac{1}{K}-1\right)}{\varepsilon_0 A}\)
Since a metal sheet of thickness \(d / 2\) is introduced, hence here, \(t=d / 2\), \(K=\propto\) (for metals)
or \(\frac{1}{K}=0\)

Taking ratio of equation (2) and (1),
\(\therefore \frac{C^{\prime}}{C}=\frac{\frac{2 \varepsilon_0 A}{d}}{\frac{\varepsilon_0 A}{d}}=\frac{2}{1}=2: 1\)