Air capacitor has capacitance ' $\mathrm{C}_1$ '. The space between two plates of capacitor is filled with…
Air capacitor has capacitance ' $\mathrm{C}_1$ '. The space between two plates of capacitor is filled with two dielectrics as shown in figure. The new capacitance of the capacitor is ' $\mathrm{C}_2$ '. The ratio $\frac{\mathrm{C}_1}{\mathrm{C}_2}$ is $(\mathrm{d}=$ distance between two plates of capacitor, $K_1$ and $K_2$ are dielectric constants of two dielectrics respectively)
For an air capacitor, $\mathrm{C}_1=\frac{\mathrm{A} \varepsilon_0}{\mathrm{~d}}$
When two dielectrics are inserted as shown, the combination acts as capacitors in series.
$\begin{aligned}
& \frac{1}{\mathrm{C}_2}=\frac{1}{\mathrm{CK}_1}+\frac{1}{\mathrm{CK}_2} \\
& \frac{1}{\mathrm{C}_2}=\frac{1}{\frac{\mathrm{~A} \varepsilon_0 \mathrm{~K}_1}{\frac{\mathrm{~d}}{2}}}+\frac{1}{\frac{\mathrm{~A} \varepsilon_0 \mathrm{~K}_2}{\frac{d}{2}}}=\frac{\mathrm{d}}{2 \varepsilon_0 \mathrm{~A}}+\left(\frac{1}{\mathrm{~K}_1}+\frac{1}{\mathrm{~K}_2}\right)
\end{aligned}$
$\begin{aligned}
C_2 & =\frac{2 \varepsilon_0 A}{d}+\left(\frac{K_1 K_2}{K_1+K_2}\right)=\frac{\frac{A \varepsilon_0}{d}}{\frac{2 A \varepsilon_0 A}{d}\left(\frac{K_1 K_2}{K_1+K_2}\right)} \\
& =\frac{K_1+K_2}{2 K_1 K_2}
\end{aligned}$
