A $$parallel-plate$$ capacitor of area A, plate separation d and capacitance C is filled with four dielectric materials having dielectric constants $$k1$$, $$k2$$,$$k3$$ and $$k4$$ as shown in the figure below. If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constant k is given by

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Infinity Learn · Accepted Answer

$$c1=2k1$$ε$$0A3d$$, $$c2=2k2$$ε$$0A3d$$, $$c3=2k3$$ε$$0A3d$$,if $$c=$$ε$$0Ad$$ $$C1=2k1c3$$, $$C2=2k2c3$$, $$C3=2k3c3Since$$ $$c1$$,$$c2$$ ,$$c3$$ are parallel ; cp $$=$$ $$c1$$ $$+$$ $$c2$$ $$+$$ $$c3=2c3k1+k2+k3$$ ; $$c4=2k4$$ε$$0Ad=2k4cSince$$ cp and $$c4$$ are series $$1cs=1cp+1c41kc=32ck1+k2+k3+12k4c$$ $$2k=3k1+k2+k3+1k4$$

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