A parallel plate capacitor has plates of area A separated by distance ‘d’ between them. It is filled with a dielectric which has dielectric constant that varies as `K (y) = k _(0) (1+ alpha y)` where ‘y’ is the vertical distance measured from base of the plates. The total capacitance of the system is best given by: `(K _(0)` is constant)

Options

A. `(in_0AK_0)/d(1+(betaA)/(2))`
B. `(in_0AK_0)/d(sqrtA+(betasqrtA)/(2))`
C. `(in_0AK_0)/d(1+(betasqrtA)/(2))`
D. `(in_0sqrt(AK_0))/d(sqrtA+(betasqrtA)/(2))`

Answer

C

Explanation

`dC=(in_(0)(sqrt(A)dy)k(y))/(d), dC=(sqrt(A)in_(0)K_(0)))(tay)dy)/(d)`
`therefore C=(sqrt(A)in_(0)K_(0))/(d) Int_(0)^(sqrt(A)) (1+betay)dy`
`=(sqrt(A)in_(0)K_(0))/(d)(sqrt(A)+(betaA)/(2)) implies (in_(0)AK_(0))/(d)(1+(betasqrt(A))/(2))`.