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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

`(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)`

Text Solution

Verified by Experts

The correct Answer is:
B
`C_(1)=(k_(1)((A)/(2))epsilon_(0))/((d//2))=(k_(1)epsilon_(0)A)/(d),C_(2)=(k_(2)epsilon_(0)A//2)/((d//2))=(k_(2)epsilon_(0)A)/(d)`
`C_(3)=(k_(3)epsilon_(0)A)/((d//2))=(2k_(3)epsilon_(0)A)/(d)`
`(1)/(C_(eq))=(1)/(C_(1)+C_(2))+(1)/(C_(3))=(1)/((epsilon_(0)A)/(d)(k_(1)+k_(2)))+(1)/((epsilon_(0)A)/(d)xx2k_(3))`
`(1)/(C_(eq))=(d)/(Aepsilon_(0))((1)/(k_(1)+k_(2))+(1)/(2k_(3)))`
`implies C_(eq)[(1)/(k_(1)+k_(2))+(1)/(2k_(3))]^(-1)(epsilon_(0)A)/(d)=0`
`k_(eq)=[(1)/(k_(1)+k_(2))+(1)/(2k_(3))]^(-1)implies (1)/(k_(eq))=(1)/(k_(1)+k_(2))+(1)/(2k_(3))`
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