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

A

(a) `1/K=1/K_1+1/K_2+1/(2K_3)`

B

(b) `1/K=(1)/(K_1+K_2)+1/(2K_3)`

C

(c) `K=(K_1K_2)/(K_1+K_2)+2K_3`

D

(d) `K=K_1+K_2+2K_3`

Text Solution

Verified by Experts

The correct Answer is:
B

Let `C_1=` Capacity of capacitor with `K_1`
`C_2=Capacity of capacitor with `K_2`
`C_3=` Capacity of capacitor with `K_3`
`:.` `C_1=K_1(A/2)(epislon_0xx2)/(4)=(Aepislon_0K_1)/(d)`
`:.` `C_2=K_2(A/2)(epislon_0xx2)/(d)=(Aepislon_0K_2)/(d)`
`:.` `C_3=K_3(A)(epislon_0xx2)/(d)=(2Aepislon_0K_3)/(d)`
`C_1` and `C_2` are in parallel
`:.` `C_(eq)=(Aepislon_0)/(d)(K_1+K_2)`
`C_(eq)` and `C_3` are in series
`:.` `1/C=(d)/(Aepislon_0(K_1+K_2))+(d)/(2Aepislon_0K_3)`
But `C=(KAepislon_0)/(d)` for isngle equivalent capacitor
`:.` `(d)/(KAepislon_0)=(d)/(Aepislon_0(K_1+K_2))+(d)/(2Aepislon_0K_3)`
or `1/K=(1)/(K_1+K_2)+(1)/(2K_3)`.
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