A parallel plate capacitor of area `A`, plate separation d` `and capacitance `C` is filled with three different dielectric materials having dielectric constants `k_1`, `k_2` and `k_3` as shown. If a single dielectric material is to be used to have the same capacitance `C` in this capacitor, then its dielectric constant `k` is given by

Options

A. `(1)/(k) = (1)/(k_(1)) + (1)/(k_(2)) + (1)/(2k_(3))`
B. `(1)/(k) = (1)/(k_(1) + k_(2)) + (1)/(2k_(3))`
C. `k = (k_(1)k_(2))/(k_(1) + k_(2)) + 2k_(3)`
D. `k = k_(1) + k_(2) + 2k_(3)`

Answer

B

Explanation

`C_(1) = (K_(1)epsilon_(0)(A)/(2))/(((d)/(2))) = (K_(1)epsilon_(0)A)/(d)`
`C_(2) = (K_(2)epsilon_(0)(A)/(2))/(((d)/(2))) = (K_(2) epsilon_(0) A)/(d)` and `C_(3) = (K_(3)epsilon_(0)(A)/(2))/(((d)/(2))) = (2K_(3)epsilon_(0)A)/(d)`
`= (1)/(C_(eq)) = (1)/(C_(1) + C_(2)) + (1)/(3) = (1)/((epsilon_(0)A)/(d)(K_(1) + K_(2))) + (1)/((epsilon_(0))/(d) xx2K_(3))`
`(1)/(C_(eq) = (d)/(epsilon_(0)A) [(1)/(K_(1) + K_(2)) + (1)/(2K_(3))]`
`C_(eq) = [(1)/(K_(1) + K_(2)) + (1)/(2K_(3))]^(-1). (epsilon_(0)A)/(d)`
So `K_(eq) = [(1)/(K_(1) + K_(2)) + (1)/(2K_(3))]^(-1)`