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A parallel plate capacitor of area A, pl...

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 isngle dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectic constant k is given by

A

`(1)/(K) =(1)/(K_(1)) +(1)/(K_(2)) +(1)/(2K_(3))`

B

`(1)/(K) = (1)/(K_(1)+K_(2)) +(1)/(2K_(3))`

C

`(1)/(K) = (K_(1)K_(2))/(K_(1)+K_(2)) +2K_(3)`

D

`K = (K_(1)K_(3))/(K_(1)+K_(3)) +(K_(2)K_(3))/(K_(2)+K_(3))`

Text Solution

Verified by Experts

The correct Answer is:
D

`C = (in_(0)A)/(d) (K_(1)K_(3))/(K_(1)+K_(3)) +(in_(0)A)/(d) (K_(2)K_(3))/(K_(2)+K_(3))`
`(Kin_(0)A)/(d) = (in_(0)A)/(d) [(K_(1)K_(3))/(K_(1)+K_(3))+(K_(2)K_(3))/(K_(2)+K_(3))]`
`K = (K_(1)K_(3))/(K_(1)+K_(3)) +(K_(2)K_(3))/(K_(2) +K_(3))`
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