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 $K\left(x\right)=K_0\left(1+\alpha x\right)$ where $x$ is the distance measured from one of the plates. If $\left(\alpha d\right)<<1,$ the total capacitance of the system is best given by the expression:

• Correct answer
  A

  $\frac{A{K}_0{\epsilon }_0}{d}\left(1+\frac{\alpha d}{2}\right)$
• B

  $\frac{A{K}_0{\epsilon }_0}{d}\left[1+{\left(\frac{\alpha d}{2}\right)}^2\right]$
• C

  $\frac{A{K}_0{\epsilon }_0}{d}\left(1+\frac{{\alpha }^2{d}^2}{2}\right)$
• D

  $\frac{A{K}_0{\epsilon }_0}{d}\left(1+\alpha d\right)$

Hence, the correct answer is option: 'A'.