A parallel plate capacitor has plates of area A separated by distance d between them. It is filled with a dielectric which has a dielectric constant that varies as $$K(x) = K_0(1 + \alpha x)$$ where $$x$$ is the distance measured from one of the plates. If $$(\alpha d) \ll 1$$, the total capacitance of the system is best given by the expression:

A

$$\frac{AK_0\varepsilon_0}{d}\left(1 + \frac{\alpha d}{2}\right)$$

B

$$\frac{AK_0\varepsilon_0}{d}\left[1 + \left(\frac{\alpha d}{2}\right)^2\right]$$

C

$$\frac{AK_0\varepsilon_0}{d}\left(1 + \frac{\alpha^2 d^2}{2}\right)$$

D

$$\frac{AK_0\varepsilon_0}{d}(1 + \alpha d)$$