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 + α x where x is the distance measured from one of the plates. If α d < < 1 , the total capacitance of the system is best given by the expression:

Step-by-step solution

Capacitance of element, d C = K 0 1 + α x ε 0 A d x ∫ 1 d C = ∫ 0 d d x K 0 ε 0 A 1 + α x ⇒ 1 C = 1 K 0 ε 0 A α ln 1 + α d Given α d ≪ 1 ⇒ 1 C = 1 K 0 ε 0 A α α d - α 2 d 2 2 ⇒ 1 C = d K 0 ε 0 A 1 - α d 2 ⇒ C = K 0 ε 0 A d 1 + α d 2