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. 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

Question

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. 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

Infinity Learn · Accepted Answer

Here,   $$C1=2$$ε$$0k1A3d$$,$$C2=2$$ε$$0k2A3dC3=2$$ε$$0k3A3d$$,$$C4=2$$ε$$0k4Ad$$ Given system of $$C1$$,$$C2$$,$$C3$$ and $$C4$$ can be simplified as ∴$$1CAB=1C1+C2+C3+1C4$$ Suppose, $$CAB=k$$ε$$0Ad1k$$ε$$0Ad=123$$ε$$0Adk1+k2+k3+12$$ε$$0Adk4$$⇒$$1k=32k1+k2+k3+12k4$$∴$$2k=3k1+k2+k3+1k4$$

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