Q161JEE Main 2008MCQ

A parallel plate capacitor with air between the plates has capacitance of $9 p F$ . The separation between its plates is ${ }^{'} d{ }^{'}$ . The space between the plates has dielectric constant $k_1=3$ and thickness $\;\frac{d}{3}$ while the other one has dielectric constant $k_2=6$ and thickness $\;\frac{2 d}{3}$ . Capacitance of the capacitor is now

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📖 Explanation

The given capacitance is equal to two capacitances connected in series where $C_1=\;\frac{k_1 \in _0 A}{d / 3}=\;\frac{3 k_1 \in _0 A}{d}$ $=\;\frac{3 \times 3 \in _0 A}{d}=\;\frac{9 \in _0 A}{d}$ and $\;C_2=\;\frac{k_2 \in _0 A}{2 d / 3}=\;\frac{3 k_2 \in _0 A}{2 d}$ $\;=\;\frac{3 \times 6 E_0 A}{2 d}=\;\frac{9 E_0 A}{d}$ The equivalent capacitance $C_{e q}$ is $\;\;\frac{1}{C_{e q}}=\;\frac{1}{C_1}+\;\frac{1}{C_2}$ $\;=\;\frac{d}{9 E_0 A}+\;\frac{d}{9 E_0 A}$ $\;=\;\frac{2 d}{9 E_0 A}$ $\; \therefore C_{e q}=\;\frac{9}{2} \;\frac{\epsilon _0 A}{d}=\;\frac{9}{2} \times 9 p F=40.5 p F$

Q163JEE Main 2007MCQ

A battery is used to charge a parallel plate capacitor till the potential difference between the plates becomes equal to the electromotive force of the battery. The ratio of the energy stored in the capacitor and the work done by the battery will be

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📖 Explanation

Required ratio $=\;\frac{\;\text{ Energy stored \in capacitor }\;}{\;\text{ Work done by the battery }\;}=\;\frac{\;\frac{1}{2} C V^2}{C e^2}$ where $C=$ Capacitance of capacitor $V=$ Potential difference, $e=e m f$ of battery $=\;\frac{\;\frac{1}{2} C e^2}{C e^2}=\;\frac{1}{2}$ $($ as $V=e)$

Q164JEE Main 2005MCQ

A parallel plate capacitor is made by stacking $n$ equally spaced plates connected alternatively. If the capacitance between any two adjacent plates is ${ }^{'} C^{'}$ then the resultant capacitance is

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📖 Explanation

As $n$ plates are joined, it means $(n-1)$ capacitor joined in parallel. $\therefore \;\text{ resultant capacitance }\;=(n-1) C$

Q165JEE Main 2005MCQ

A fully charged capacitor has a capacitance ${ }^{'} C^{'}$ . It is discharged through a small coil of resistance wire embedded in a thermally insulated block of specific heat capacity ${ }^{'} s^{'}$ and mass ${ }^{'} m^{'}$ . If the temperature of the block is raised by ${ }^{'} \Delta T^{'}$ , the potential difference ${ }^{'} v^{'}$ across the capacitance is

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📖 Explanation

Applying conservation of energy, $\;\;\frac{1}{2} C V^2=m . s \Delta T$ $\;V=\sqrt{\;\frac{2 m . s . \Delta T}{C}}$

Q166JEE Main 2003MCQ

A sheet of aluminium foil of negligible thickness is introduced between the plates of a capacitor. The capacitance of the capacitor

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📖 Explanation

The capacitancce of parallel plate capacitor in which a metal plate of thickness $t$ is inserted is given by $C=\;\frac{\epsilon _0 A}{d-t} .$ Here $t \to 0 \;\; \therefore C=\;\frac{\epsilon _0 A}{d}$

Q167JEE Main 2003MCQ

The work done in placing a charge of $8 \times 10^{-18}$ coulomb on a condenser of capacity 100 micro-farad is

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📖 Explanation

The work done is stored as the potential energy. The potential energy stored in a capacitor is given by $\;U=\;\frac{1}{2} \;\frac{Q^2}{C}$ $\;=\;\frac{1}{2} \times \;\frac{ (8 \times 10^{-18}) ^2}{100 \times 10^{-6}}$ $\;=32 \times 10^{-32} J$

Q168JEE Main 2002MCQ

Capacitance (in $F$ ) of a spherical conductor with radius $1 m$ is

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📖 Explanation

For an isolated sphere, the capacitance is given by $C=4 \pi E_0 r$ $=\;\frac{1}{9 \times 10^9} \times 1$ $=1.1 \times 10^{-10} {\text{F}}$

Q169JEE Main 2002MCQ

If there are $n$ capacitors in parallel connected to $V$ volt source, then the energy stored is equal to

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📖 Explanation

The equivalent capacitance of $n$ identical capacitors of capacitance $C$ is equal to $n C$ . Energy stored in this capacitor $E=\;\frac{1}{2}(n C) V^2=\;\frac{1}{2} n C V^2$