A parallel plate capacitor is formed by two plates each of area $$30\pi$$ cm$$^2$$ separated by $$1$$ mm. A material of dielectric strength $$3.6 \times 10^{7}$$ 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 $$7 \times 10^{-6}$$ C, the value of dielectric constant of the material is :
[Use $$\frac{1}{4\pi\varepsilon_0} = 9 \times 10^{9}$$ N m$$^2$$ C$$^{-2}$$]

We are given: area $$A = 30\pi$$ cm$$^2 = 30\pi \times 10^{-4}$$ m$$^2$$, separation $$d = 1$$ mm $$= 10^{-3}$$ m, dielectric strength $$E_{max} = 3.6 \times 10^7$$ V/m, maximum charge $$Q = 7 \times 10^{-6}$$ C, $$\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9$$ N m$$^2$$ C$$^{-2}$$.

From the relation $$\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9$$, it follows that

$$ \varepsilon_0 = \frac{1}{4\pi \times 9 \times 10^9} $$

The capacitance with dielectric is given by

$$ C = \frac{\kappa \varepsilon_0 A}{d} $$

The maximum voltage before dielectric breakdown is

$$ V_{max} = E_{max} \times d = 3.6 \times 10^7 \times 10^{-3} = 3.6 \times 10^4 \text{ V} $$

Using $$Q = CV$$ yields

$$ Q = C \times V_{max} = \frac{\kappa \varepsilon_0 A}{d} \times E_{max} \times d = \kappa \varepsilon_0 A \times E_{max} $$

$$ \kappa = \frac{Q}{\varepsilon_0 \times A \times E_{max}} $$

Substituting the given values:

$$ \kappa = \frac{7 \times 10^{-6}}{\frac{1}{4\pi \times 9 \times 10^9} \times 30\pi \times 10^{-4} \times 3.6 \times 10^7} $$

Computing the denominator step by step:

$$ \varepsilon_0 \times A = \frac{30\pi \times 10^{-4}}{4\pi \times 9 \times 10^9} = \frac{30 \times 10^{-4}}{4 \times 9 \times 10^9} = \frac{30 \times 10^{-4}}{36 \times 10^9} = \frac{10^{-4}}{1.2 \times 10^9} = \frac{1}{1.2} \times 10^{-13} $$

$$ \varepsilon_0 \times A \times E_{max} = \frac{1}{1.2} \times 10^{-13} \times 3.6 \times 10^7 = \frac{3.6}{1.2} \times 10^{-6} = 3 \times 10^{-6} $$

$$ \kappa = \frac{7 \times 10^{-6}}{3 \times 10^{-6}} = \frac{7}{3} \approx 2.33 $$

Therefore, the correct answer is Option D.