The gap between the plates of a… - JEE Main 2014 Physics PYQ

JEE Main 2014 — Physics Electromagnetism

2014mcqhard

The gap between the plates of a parallel plate capacitor of area $A$ and distance between plates $d$ , is filled with a dielectric whose relative permittivity varies linearly from ${\epsilon }_{1}$ at one plate to ${\epsilon }_{2}$ at the other. The capacitance of the capacitor is

Official previous-year question

Held on 19 Apr 2014 · Verified 6 Jul 2026.

Options

A
$\frac{{\epsilon }_{0}({\in }_{2}-{\in }_{1})A}{[d\mathrm{ln}({\in }_{2}/{\in }_{1})]}$
B
$\frac{{\epsilon }_{0}({\in }_{2}+{\in }_{1})A}{2d}$
C
$\frac{{\epsilon }_{0}({\in }_{1}+{\in }_{2})A}{d}$
D
$\frac{{\epsilon }_{0}A}{[dln({\in }_{2}/{\in }_{1})]}$

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Solution

$\frac{1}{dC}=\frac{dx}{{\epsilon }_{0}\epsilon A}$

$=\frac{dx}{{\epsilon }_{0}[{\epsilon }_{1}+\frac{({\epsilon }_{2}-{\epsilon }_{1})}{d}x]A}$

$\frac{1}{C}=\int \frac{1}{dC}=\frac{1}{{\in }_{0}A}\int _{0}^{x}\frac{dx}{{\in }_{1}+(\frac{{\in }_{2}-{\in }_{1}}{d})}$

$=\frac{\mathrm{ln}({\in }_{2}/{\in }_{1})d}{{\in }_{0}A({\in }_{2}-{\in }_{1})}$

$C=\frac{{\in }_{0}A({\in }_{2}-{\in }_{1})}{\mathrm{ln}({\in }_{2}/{\in }_{1})\cdot d}$

JEE Main 2014 Physics, Capacitance — solution figure

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Options

$\frac{{\epsilon }_{0}({\in }_{2}-{\in }_{1})A}{[d\mathrm{ln}({\in }_{2}/{\in }_{1})]}$

$\frac{{\epsilon }_{0}({\in }_{2}+{\in }_{1})A}{2d}$

$\frac{{\epsilon }_{0}({\in }_{1}+{\in }_{2})A}{d}$

$\frac{{\epsilon }_{0}A}{[dln({\in }_{2}/{\in }_{1})]}$

Solution

$\frac{1}{dC}=\frac{dx}{{\epsilon }_{0}\epsilon A}$

$=\frac{dx}{{\epsilon }_{0}[{\epsilon }_{1}+\frac{({\epsilon }_{2}-{\epsilon }_{1})}{d}x]A}$

$\frac{1}{C}=\int \frac{1}{dC}=\frac{1}{{\in }_{0}A}\int _{0}^{x}\frac{dx}{{\in }_{1}+(\frac{{\in }_{2}-{\in }_{1}}{d})}$

$=\frac{\mathrm{ln}({\in }_{2}/{\in }_{1})d}{{\in }_{0}A({\in }_{2}-{\in }_{1})}$

$C=\frac{{\in }_{0}A({\in }_{2}-{\in }_{1})}{\mathrm{ln}({\in }_{2}/{\in }_{1})\cdot d}$

JEE Main 2014 Physics, Capacitance — solution figure

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Solution

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Mechanics

Modern Physics

Optics

Thermodynamics

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