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A parallel -plate capacitor of area A, p...

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

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)`

Text Solution

Verified by Experts

The correct Answer is:
A
Circuit can be redrawn as

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