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 dis

Question

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:

TestHub · Accepted Answer

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

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