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)`.

Explanation

The given capacitor is equivalent to two capacitors in series. Let `C_(1)` and `C_(2)` be their capacities. Then
`(1)/(C)=int [(1)/(dC_(1))+(1)/(dC_(2))]`
Consider an element of width `dx` at a distance `x` from the left plate. Then
`dC_(1)=((epsilon_(0)+kx)A)/(dx) for 0 ltx lt (d)/(2)`
and `dC_(2)=({in_(0)+k(d-x)}A)/(dx) for (d)/(2) lt x le d`
on substituting these two values we get
`(1)/(C)-int(1)/(dC)=(2)/(KA) l n((2 in_(0)+ Kd)/(2 in_(0))) rArr C =(KA)/(2)l n((2 in_(0)+Kd)/(2 in_(0)))`.