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QJEEM21Ph72da: JEE Main 2021 Physics - Class 12 JEEM21Ph72da Solution: A parallel plate capacitor with plate area 'A' and distance of…

A parallel plate capacitor with plate area 'A' and distance of separation 'd' is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as :

ε(x)=ε0+kx\varepsilon (x) = {\varepsilon _0} + kx , for \left( {0 < x \le {d \over 2}} \right)

ε(x)=ε0+k(d−x)\varepsilon (x) = {\varepsilon _0} + k(d - x) , for (d2≤x≤d)\left( {{d \over 2} \le x \le d} \right)

Multiple Choice Options

  • A (ε0+kd2)2/kA{\left( {{\varepsilon _0} + {{kd} \over 2}} \right)^{2/kA}}
  • B kA2ln⁡(2ε0+kd2ε0){{kA} \over {2\ln \left( {{{2{\varepsilon _0} + kd} \over {2{\varepsilon _0}}}} \right)}}
  • C 0
  • D kA2ln⁡(2ε02ε0−kd){{kA} \over 2}\ln \left( {{{2{\varepsilon _0}} \over {2{\varepsilon _0} - kd}}} \right)

Step-by-Step Verified Solution

JEE Main 2021 (Online) 25th July Morning Shift Physics - Capacitor Question 82 English Explanation

Taking an element of width dx at a distance x(x < d/2) from left plate

dc=(ε0+kx)Adxdc = {{({\varepsilon _0} + kx)A} \over {dx}}

Capacitance of half of the capacitor

1C=∫0d/21dc=1A∫0d/2dxε0+kx{1 \over C} = \int\limits_0^{d/2} {{1 \over {dc}} = {1 \over A}\int\limits_0^{d/2} {{{dx} \over {{\varepsilon _0} + kx}}} }

1C=1kAln⁡(ε0+kd/2ε0){1 \over C} = {1 \over {kA}}\ln \left( {{{{\varepsilon _0} + kd/2} \over {{\varepsilon _0}}}} \right)

Capacitance of second half will be same

Ceq=C2=kA2ln⁡(2ε0+kd2ε0){C_{eq}} = {C \over 2} = {{kA} \over {2\ln \left( {{{2{\varepsilon _0} + kd} \over {2{\varepsilon _0}}}} \right)}}

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