Three different dielectrics are filled in a parallel plate capacitor as shown. What should be the dielectric constant of a material, which when fully filled between the plates produces same capacitance?
Medium
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(c)
$C_1=\frac{6 A \varepsilon_0}{2 d}=\frac{3 A \varepsilon_0}{d}$
$C_2=\frac{A \varepsilon_0}{2\left(d-\frac{d}{2}\left(1-\frac{1}{3}\right)-\frac{d}{2}\left(1-\frac{1}{6}\right)\right)}$
$C_2=\frac{A \varepsilon_0}{2\left(\frac{d}{6}+\frac{d}{12}\right)}$
$C_2=\frac{A \varepsilon_0}{2\left(\frac{d}{4}\right)}$
$C_2=\frac{2 A \varepsilon_0}{d}$
$C=C_1+C_2=\frac{5 A \varepsilon_0}{d}$
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