Capacitors

JEE Main level, single correct, medium

73 of 180

Question 73 of 180

A parallel plate capacitor has capacitance CC, when there is vacuum within the parallel plates. A sheet having thickness (13)rd\left(\frac{1}{3}\right)^{\mathrm{rd}} of the separation between the plates and relative permittivity KK is introduced between the plates. The new capacitance of the system is :

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Answer: (C) 3KC2K+1\frac{3 K C}{2 K+1}

Original capacitance: C=ε0AdC = \frac{\varepsilon_0 A}{d}
After inserting dielectric of thickness d3\frac{d}{3}, system becomes two capacitors in series.
Vacuum region: C1=ε0A2d/3=3C2C_1 = \frac{\varepsilon_0 A}{2d/3} = \frac{3C}{2}
Dielectric region: C2=Kε0Ad/3=3KCC_2 = \frac{K\varepsilon_0 A}{d/3} = 3KC
Equivalent capacitance: 1Cnew=1C1+1C2=23C+13KC=2K+13KC\frac{1}{C_{new}} = \frac{1}{C_1} + \frac{1}{C_2} = \frac{2}{3C} + \frac{1}{3KC} = \frac{2K+1}{3KC}
Cnew=3KC2K+1C_{new} = \frac{3KC}{2K+1}

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