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Question
A slab of material of dielectric constant K has the same area as the plates of a parallel plate capacitor but has a thickness \[\frac{3d}{4}\]. Find the ratio of the capacitance with dielectric inside it to its capacitance without the dielectric.
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Solution
\[\text { Capacitance of a capacitor without dielectric is given by }: \]
\[ C_o = \frac{\epsilon_o A}{d} . . . . . (i)\]
\[\text { Capacitance of capacitor when its plates are partly filled with dielectric of thickness t and of same area as the plates is given by }: \]
\[C = \frac{\epsilon_o A}{d - t (1 - \frac{1}{K})}\]
\[\text { Here} , t = \frac{3d}{4}\]
\[C = \frac{\epsilon_o A}{d - \frac{3d}{4}(1 - \frac{1}{K})} = \frac{\epsilon_o A}{\frac{dK + 3d}{4K}} = \frac{\epsilon_o A(4K)}{dK + 3d}\]
\[ = \frac{\epsilon_o A(4K)}{d(K + 3)} = \frac{4K}{(K + 3)} \times \frac{\epsilon_o A}{d}\]
\[\text { Therefore, the ratio of the capacitance with dielectric inside it to its capacitance without the dielectric is }\]
\[\frac{C_o}{C} = \frac{\frac{4K}{(K + 3)} \times \frac{\epsilon_o A}{d}}{\frac{\epsilon_o A}{d}} = \frac{4K}{(K + 3)}\]
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