CapacitanceHard

Question

A parallel plate capacitor has capacitance C , when there is vacuum within the parallel plates.

A sheet having thickness $\left( \frac{1}{3} \right)^{rd}$ of the separation between the plates and relative permittivity K is introduced between the plates. The new capacitance of the system is :

Options

A.$\frac{3KC}{2\text{ }K + 1}$
B.$\frac{\text{~CK~}}{2 + \text{~K~}}$
C.$\frac{3{CK}^{2}}{(2\text{ }K + 1)^{2}}$
D.$\frac{4KC}{3\text{ }K - 1}$

Solution

$${C_{1} = \frac{3\text{ }A\epsilon_{0}}{2\text{ }d},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C_{2} = \frac{3\text{ }A \in_{0} \times K}{d} }{C_{1} = \frac{3}{2}C\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C_{2} = 3KC }{Ceq = \frac{C_{1}C_{2}}{C_{1} + C_{2}} = \frac{\frac{3}{2}C \times 3KC}{\frac{3}{2}C + 3KC} }{Ceq = \frac{\frac{9}{2}{KC}^{2}}{\frac{3}{2}C(2\text{ }K + 1)} = \frac{3KC}{2\text{ }K + 1}}$$

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