A parallel plate capacitor is made of two square plates of side ' $a$ ', separated by a distance $\mathrm{d}(\mathrm{d} \ll a$ ). The lower triangular portion is filled with a dielectric of dielectric constant K, as shown in the figure. Capacitance of this capacitor is

(B) $\frac{\mathrm{K} \varepsilon_{0} \mathrm{a}^{2}}{\mathrm{~d}(\mathrm{~K}-1)} \ln \mathrm{K}$

(A) $\frac{\mathrm{K} \varepsilon_{0} \mathrm{a}^{2}}{2 \mathrm{~d}(\mathrm{~K}+1)}$

(C) $\frac{K \varepsilon_{0} a^{2}}{d} \ln K$

(D) $\frac{1}{2} \frac{\mathrm{~K} \varepsilon_{0} \mathrm{a}^{2}}{\mathrm{~d}}$

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Capacitance

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. Capacitance of this capacitor is

Answer: (B)

Slice the capacitor into thin vertical strips of width

d x

at distance

x

from the thin end, where the dielectric height is

y = \frac{d}{a} x

. Each strip is an air gap in series with a dielectric-filled gap, so

d C = \frac{K ε _{0} a d x}{K d + ( 1 - K ) y}

. Substituting

y = \frac{d}{a} x

and integrating from

0

a

gives

C = \frac{K ε _{0} a ^{2}}{d ( K - 1 )} ln K

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Single correct option