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Calculate the capacitance of a parallel ...

Calculate the capacitance of a parallel plate capacitor, with plate area `A` and distance between the plates `d`, when filled with a dielectric whose permittivity varies as
`in(x)=in_(0)+kx(0 lt x lt (d)/(2)), in(x)=in_(0) +k(d-x)((d)/(2) lt x le d)`.

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
`(1)/(C_(1)) = int (1)/(dC_(1)) = overset(d//2)underset(0)int (dx)/((in_(0)+betax)A)`
`= (1)/(betaA) [l n(in_(0)+betax)]_(0)^(d//2)`
`(1)/(C_(1))=(1)/(betaA)[l n(in_(0)+(betad)/(2))-l nin_(0)]`
`= (1)/(betaA) ln ((2in_(0)+betad)/(2in_(0)))`
`(1)/(C_(2)) = int (1)/(dC_(2)) = (1)/(A) overset(d)underset(d//2)int (dx)/(in_(0)+beta(d-x))`
`=- (1)/(betaA) [l n{in_(0)+beta(d-x)}]_(d//2)^(d)`
`(1)/(C_(1))=(1)/(betaA)[l n{in_(0)+beta(d-d)}]-l n[in_(0)+(betad)/(2)]`
`=- (1)/(betaA) [l nin_(0) -l n(2in_(0) +betad)/(2)]`
`(1)/(C_(2)) =- (1)/(betaA) ln (2in_(0))/(2in_(0)+betad)`
`(1)/(C_(2)) = (1)/(betaA) l n (2in_(0) +betad)/(2in_(0))`
`(1)/(C ) = (1)/(C_(1))+(1)/(C_(2)) =(1)/(betaA)[2l n(2in_(0)+betad)/(2in_(0))]`
`C =(betaA//2)/(l n(2in_(0)+betad)/(2in_(0)))`
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