Consider a parallel plate capacitor with plates in the spape of square and in XY-plane. The gap between the plates is filled with dielectric material. The dielectric constant k varies with X-axis as `k (x) = [1+ ((x)/(L)) ^(alpha )],` where `alpha ` is a constant. Let `D_(d) and C_(a)` are capacitance in the presence of dielectric and air, respectively. If the ratio `(C_(d))/(C_(a)) = 7/6, ` then the value of `alpha` must be

Options

A. 3
B. 5
C. 7
D. 9

Answer

B

Explanation

Area of square plate, ` A= L ^(2)`
Consider an elemental capacitor at distance x from above of thickness dx. Capacitance of elemental capacitor,
`dC=(k epsi_(0)dA)/(d) = (k epsi_(0) (L. dx))/(d)`
All such elemental capacitors will be in parallel and equivalent capacitance will be sum of all, so
`C_(d) = (epsi _(0) L )/(d ) int _(0) ^(L ) 1 + ((X)/(L)). dx`
`C _(d) = (epsi _(0) L)/(d) [(x + ((X)/(L))^(1+ alpha ))/((1+omega). (1)/(L))]_(0) ^(L)`
`C_(d) = (epsi _(0)L)/(d ) [L+ (L)/(1 + alpha )] implies C _(d) = (epsi _(0) L ^(2))/(d) ((2+alpha )/(1 + alpha ))`
It is given, `(C_(d))/(C_(a)) = 7/6`
where, `C _(a),` capacitance in absence of dielectric.
`implies ((epsi _(0)L^(2))/(d ) ((2 + alpha )/(1+ alpha)))/((epsi _(0) L ^(2))/(d )) = 7/6 implies (2+ alpha )/(1+ alpha) = 7/6`
`implies 12 + 6 alpha = 7+ 7 alpha implies alpha =5`