The space between the plates of a parallel plate capacitor of plate area $4\ \text{cm}^2$ and separation $d = 1.77\ \text{mm}$ is filled with two uniform dielectric layers, each of thickness $d/2$ and covering the whole plate area, with dielectric constants $5$ and $3$. Another capacitor of capacitance $7.5\ \text{pF}$ is connected in parallel with it. The effective capacitance of this combination is ______ pF. (Given $\varepsilon_0 = 8.85\times10^{-12}\ \text{F/m}$)
Numerical value type. Enter your answer.
Answer: 15
Each layer is a capacitor of gap $d/2$:
$$\frac{\varepsilon_0A}{d/2} = \frac{8.85\times10^{-12}\times4\times10^{-4}}{0.885\times10^{-3}} = 4\ \text{pF}$$
So $C_1 = 5\times4 = 20\ \text{pF}$ and $C_2 = 3\times4 = 12\ \text{pF}$, in series:
$$C = \frac{20\times12}{20 + 12} = 7.5\ \text{pF}$$
Adding the parallel $7.5\ \text{pF}$: $C_{\text{eff}} = 15\ \text{pF}$.
Solution by Sreeraj P, M.Sc Physics