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_{2}) (K_{3} + K_{4})}{(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_{3}) (K_{2} + K_{4})}{K_{1} + K_{2} + K_{3} + K_{4}}

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Solution

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

$C_{12} = \frac{C_{1} C_{2}}{C_{1} + C_{2}} = \frac{\frac{k_{1} \in_{0} \frac{L}{2} \times L}{d / 2} . \frac{k_{2} [\in_{0} \frac{L}{2} \times L]}{d / 2}}{(k_{1} + K_{2}) ⎡ ⎢ ⎢ ⎢ ⎣ \frac{\in_{0} . \frac{L}{2} \times L}{d / 2} ⎤ ⎥ ⎥ ⎥ ⎦}$

$C_{12} = \frac{k_{1} k_{2}}{k_{1} + k_{2}} \frac{\in_{0} L^{2}}{d}$

in the same way we get , $C_{34} = \frac{k_{3} k_{4}}{k_{3} + k_{4}} \frac{\in_{0} L^{2}}{d}$

$∴ C_{e q} = C_{12} + C_{34} = [\frac{k_{1} k_{2}}{k_{1} + k_{2}} + \frac{k_{3} k_{4}}{k_{3} + k_{4}}] \frac{\in_{0} L^{2}}{d}$

Now if $k_{e q} - k, C_{e q} = \frac{k \in_{0} L^{2}}{d}$

on comparing equation (i) to equation (ii) , we get

$k_{e q} = \frac{k_{1} k_{2} (k_{3} + k_{4}) + k_{3} K_{4} (k_{1} + k_{2})}{(k_{1} + k_{2}) (k_{3} + k_{4})}$

This does not match with any of the options so probably they have assumed the wrong combination

$C_{13} = \frac{k_{1} \in_{0} L \frac{L}{2}}{d / 2} + k_{3} \in_{0} \frac{L . \frac{L}{2}}{d / 2}$

$= (k_{1} + k_{3}) \frac{\in_{0} L^{2}}{d}$

$C_{24} = (k_{2} + k_{4}) \frac{\in_{0} L^{2}}{d}$

$C_{e q} = \frac{C_{13} C_{24}}{C_{13} C_{24}} = \frac{(K_{1} + K_{3}) (k_{2} + k_{4})}{(k_{1} + k_{2} + k_{3} + k_{4})} \frac{\in_{0} L^{2}}{d}$

$= \frac{k \in_{0} L^{2}}{d}$

$k = \frac{(k_{1} + k_{3}) (k_{2} + k_{4})}{(k_{1} + k_{2} + k_{3} + k_{4})}$

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Then $K_{e} q = ?$

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A parallel plate capacitor with square plates is filled with four dielectrics of dielectric constants K1,K2,K3,K4 arranged as shown in the figure. The effective dielectric constant K will be :

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 :