A parallel plate capacitor with air as a dielectric has capacitance $C$ . A slab of dielectric constant $K$ , having same thickness as the separation between the plates is introduced so as to fill one-fourth of the capacitor as shown in the figure. The new capacitance will be

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The two condensers with K and with air are in parallel. With air, C 1 ​ = d ε 0 ​ ​ ( 4 3 A ​ ) = 4 d 3 ε 0 ​ A ​ With medium, C 2 ​ = d ε 0 ​ K ​ ( 4 A ​ ) = 4 d ε 0 ​ A K ​ ∴ C ′ = C 1 ​ + C 2 ​ or C ′ = 4 d 3 ε 0 ​ A ​ + 4 d ε 0 ​ A K ​ = d ε 0 ​ A ​ [ 4 3 ​ + 4 K ​ ] or C ′ = 4 C ​ ( K + 3 ) [ ∵ C = d A ε 0 ​ ​ ]

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