A parallel plate capacitor is formed by two plates each of area $30\pi\ \text{cm}^2$ separated by $1\ \text{mm}$. A material of dielectric strength $3.6\times10^7\ \text{V m}^{-1}$ is filled between the plates. If the maximum charge that can be stored on the capacitor without causing any dielectric breakdown is $7\times10^{-6}\ \text{C}$, the value of dielectric constant of the material is: [Use $\dfrac{1}{4\pi\varepsilon_0} = 9\times10^9\ \text{N m}^2\text{C}^{-2}$]
Answer: (D) 2.33
The field inside the dielectric is $E = \dfrac{Q}{K\varepsilon_0 A}$, and it must not exceed the dielectric strength:
$$K = \frac{Q}{\varepsilon_0 A E_{\max}}$$
With $\varepsilon_0 = \dfrac{1}{36\pi\times10^9}$ and $A = 30\pi\times10^{-4}\ \text{m}^2$: $\varepsilon_0 A = \dfrac{30}{36}\times10^{-13}$.
$$K = \frac{7\times10^{-6}}{\frac{30}{36}\times10^{-13}\times3.6\times10^{7}} = \frac{7\times10^{-6}}{3\times10^{-6}} \approx 2.33$$
Solution by Sreeraj P, M.Sc Physics