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Capacitor question

2016 · Q120

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Capacitor
2016 · Q120

Capacitor question

2016 · Q120

NEETPhysicsCapacitorMCQ+4 / −1

A parallel-plate capacitor of area a, plate separation d and capacitance C is filled with four dielectric materials having dielectric constants k₁, k₂, k₃ and k₄ as shown in the figure. If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constant k is given by
NEET 2016 Phase 2 Physics - Capacitor Question 35 English

NEET 2016 Phase 2 Physics - Capacitor Question 35 English

A
k = k₁ + k₂ + k₃ + 3k₄
B
k = ${2 \over 3}$ (k₁ + k₂ + k₃) + 2k₄
C
${2 \over k}$ = ${3 \over {{k_1} + {k_2} + {k_3}}} + {1 \over {{k_4}}}$
D
${1 \over k}$ = ${1 \over {{k_1}}} + {1 \over {{k_2}}} + {1 \over {{k_3}}} + {3 \over {2{k_4}}}$

View written solutionFree

Correct answer: C

Here, ${C_1} = {{2{\varepsilon _0}{k_1}A} \over {3d}},{C_2} = {{2{\varepsilon _0}{k_2}A} \over {3d}}$

${C_3} = {{2{\varepsilon _0}{k_3}A} \over {3d}},{C_4} = {{2{\varepsilon _0}{k_4}A} \over d}$

Given system of C₁, C₂, C₃ and C₄ can be simplified as

NEET 2016 Phase 2 Physics - Capacitor Question 35 English Explanation

\therefore

{1 \over {{C_{AB}}}} = {1 \over {{C_1} + {C_2} + {C_3}}} + {1 \over {{C_4}}}

Suppose,

{C_{AB}} = {{k{\varepsilon _0}A} \over d}

{C_{AB}} = {1 \over {k\left( {{{{\varepsilon _0}A} \over d}} \right)}} = {1 \over {{2 \over 3}{{{\varepsilon _0}A} \over d}\left( {{k_1} + {k_2} + {k_3}} \right)}} + {1 \over {{{2{\varepsilon _0}A} \over d}{k_4}}}

\Rightarrow {1 \over k} = {1 \over {2\left( {{k_1} + {k_2} + {k_3}} \right)}} + {1 \over {2{k_4}}}

\therefore {2 \over k} = {3 \over {{k_1} + {k_2} + {k_3}}} + {1 \over {{k_4}}}

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