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Question:

A parallel plate capacitor is made of two square plates of side 'a', separated by a distance d (d<<a). The lower triangular portion is filled with a dielectric of dielectric constant K, as shown in the figure. The capacitance of this capacitor is :

kε₀a²/2d(K+1)

1/2kε₀a²/d

kε₀a²/dlnK

kε₀a²/d(K½)lnK

Solution:

Let's consider a strip of thickness dx at a distance of x from the left end as shown in the figure.
y/x = a/d ⇒ y = (a/d)x
C₁ = ε₀a dx/d-y
and C₂ = kε₀a dx/y
Ceq = C₁.C₂/(C₁+C₂) = kε₀a dx/[kd + (1-k)y]
On integrating it from 0 to a, we will get
kε₀a²/d(K½)lnK