A parallel -plate capacitor of area `A`, plate separation `d` and capacitance `C` is filled with four dielectric materials having dielectric constant `k_(1), k_(2), k_(3)` and `k_(4)` as shown in the figure below. If a single dielectric materical is to be used to have the same capacitance `C` in this capacitor, then its dielectric constant `k` is given by

Options

A. `(2)/(k) = (3)/(k_(1) + k_(2) + k_(3)) + (1)/(k_(4))`
B. `(1)/(k) = (1)/(k_(1) + (1)/(k_(2)) + (1)/(k_(3)) + (3)/(2k_(4))`
C. `k = k_(1) + k_(2) + k_(3) + 3k_(4)`
D. `k = (2)/(3) (k_(1) + k_(2) + k_(3)) + 2k_(4)`

Answer

A

Explanation

Circuit can be redraws as

`(1)/(C ) = (1)/(C_(1)) + (1)/(C_(2))`
`(d)/(A in_(0)k) = (1)/((A)/(3)in_(0) ((k_(1)+k_(2)+k_(3)))/(d//2))+(1)/((Ain_(0)k_(4))/(d//2))`
`rArr (2)/(k) = (3)/(k_(1)+k_(2)+(k_(3)) + (1)/(k_(4))`