Two parallel plate air capacitors of same capacity ' C ' are connected in series to a battery of emf ' \(E\) '. Then one of the capacitors is completely filled with dielectric material of constant ' \(K\) '. The change in the effective capacity of the series combination is

\(\begin{aligned} \quad \frac{1}{\mathrm{C}_1} & =\frac{1}{\mathrm{C}}+\frac{1}{\mathrm{C}}=\frac{2}{\mathrm{C}} \\ \therefore \quad \mathrm{C}_1 & =\frac{\mathrm{C}}{2} \\ \frac{1}{\mathrm{C}_2} & =\frac{1}{\mathrm{C}}+\frac{1}{\mathrm{KC}} \\ \frac{1}{\mathrm{C}_2} & =\frac{1}{\mathrm{C}}\left[1+\frac{1}{\mathrm{~K}}\right] \\ \therefore \quad \mathrm{C}_2 & =\frac{\mathrm{CK}}{\mathrm{K}+1} \\ \therefore \mathrm{C} & =\frac{\mathrm{CK}}{\mathrm{K}+1}-\frac{\mathrm{C}}{2}=\mathrm{C}\left[\frac{\mathrm{K}}{\mathrm{K}+1}-\frac{1}{2}\right] \\ \therefore \quad \Delta \mathrm{C} & =\frac{\mathrm{C}}{2}\left[\frac{\mathrm{~K}-1}{\mathrm{~K}+1}\right]\end{aligned}\)