Capacitors

JEE Main level, numerical answer, medium

101 of 180

Question 101 of 180

In a parallel plate capacitor set up, the plate area of capacitor is 2m22{m}^{2} and the plates are separated by 1m1m. If the space between the plates are filled with a dielectric material of thickness 0.5m0.5m and are 2m22{m}^{2} (see figure) the capacitance of the set-up will be ϵ0{\epsilon }_{0} ________.
(Dielectric constant of the material =3.2=3.2)
(Round off to the Nearest Integer)

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Answer: 3

The given system can be considered as a series combination of two capacitors. The equivalent capacitance can be given as,
1C=1C1+1C2⇒1C=d2Kϵ0A+d2ϵ0A\dfrac{1}{C}=\dfrac{1}{{C}_{1}}+\dfrac{1}{{C}_{2}} \Rightarrow \dfrac{1}{C}=\dfrac{\dfrac{d}{2}}{K{\epsilon }_{0}A}+\dfrac{\dfrac{d}{2}}{{\epsilon }_{0}A}
⇒C=ϵ0Ad2K+d2=2ϵ0AdK+d\Rightarrow C=\dfrac{{\epsilon }_{0}A}{\dfrac{d}{2K}+\dfrac{d}{2}}=\dfrac{2{\epsilon }_{0}A}{\dfrac{d}{K}+d}
=2×2ϵ013.2+1=4×3.24.2ϵ0=\dfrac{2\times 2{\epsilon }_{0}}{\dfrac{1}{3.2}+1}=\dfrac{4\times 3.2}{4.2}{\epsilon }_{0}
=3.04ϵ0=3.04{\epsilon }_{0}

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