A parallel plate capacitor with square plates is filled with four dielectrics of dielectric constants $K_{1}, K_{2}, K_{3}, K_{4}$ arranged as shown in the figure. The effective dielectric constant $K$ will be

K = \frac{(K_{1} + K_{2}) (K_{3} + K_{4})}{2 (K_{1} + K_{2} + K_{3} + K_{4})}

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K = \frac{(K_{1} + K_{4}) (K_{2} + K_{3})}{2 (K_{1} + K_{2} + K_{3} + K_{4})}

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K = \frac{(K_{1} + K_{2}) (K_{3} + K_{4})}{K_{1} + K_{2} + K_{3} + K_{4}}

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K = \frac{(K_{1} + K_{3}) (K_{2} + K_{4})}{K_{1} + K_{2} + K_{3} + K_{4}}

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Solution

The correct option is D $K = \frac{(K_{1} + K_{3}) (K_{2} + K_{4})}{K_{1} + K_{2} + K_{3} + K_{4}}$

As $C_{1}$ and $C_{3}$ are in parallel,
$C_{13} = C_{1} + C_{3} = (K_{1} + K_{3}) ⎡ ⎢ ⎢ ⎢ ⎣ \frac{ϵ_{0} \frac{L}{2} \times L}{d / 2} ⎤ ⎥ ⎥ ⎥ ⎦$
$C_{24} = C_{4} + C_{2} = (K_{4} + K_{2}) ⎡ ⎢ ⎢ ⎢ ⎣ \frac{ϵ_{0} \frac{L}{2} \times L}{d / 2} ⎤ ⎥ ⎥ ⎥ ⎦$

As combination of $C_{13} & C_{24}$ are in series
$C_{e q} = \frac{C_{13} C_{24}}{C_{13} + C_{24}} = \frac{(K_{1} + K_{3}) ⎡ ⎢ ⎢ ⎢ ⎣ \frac{ϵ_{0} \frac{L}{2} \times L}{d / 2} ⎤ ⎥ ⎥ ⎥ ⎦ (K_{4} + K_{2}) ⎡ ⎢ ⎢ ⎢ ⎣ \frac{ϵ_{0} \frac{L}{2} \times L}{d / 2} ⎤ ⎥ ⎥ ⎥ ⎦}{(K_{1} + K_{3}) ⎡ ⎢ ⎢ ⎢ ⎣ \frac{ϵ_{0} \frac{L}{2} \times L}{d / 2} ⎤ ⎥ ⎥ ⎥ ⎦ + (K_{4} + K_{2}) ⎡ ⎢ ⎢ ⎢ ⎣ \frac{ϵ_{0} \frac{L}{2} \times L}{d / 2} ⎤ ⎥ ⎥ ⎥ ⎦}$
$C_{e q} = \frac{(K_{1} + K_{3}) (K_{2} + K_{4})}{K_{1} + K_{2} + K_{3} + K_{4}} \frac{ϵ_{0} L^{2}}{d} \dots (i)$

Now if $C_{e q} = \frac{K_{e q} ϵ_{0} L^{2}}{d} . . . . (i i)$

on comparing equation $(i)$ to equation $(i i)$ , we get
$K_{e q} = \frac{(K_{1} + K_{3}) (K_{2} + K_{4})}{K_{1} + K_{2} + K_{3} + K_{4}}$

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