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Q.

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

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a

k=k1+k2+k3+3k4

b

k=23k1+k2+k3+2k4

c

2k=3k1+k2+k3+1k4

d

1k=1k1+1k2+1k3+32k4

answer is C.

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Detailed Solution

Here,   C1=2ε0k1A3d,C2=2ε0k2A3dC3=2ε0k3A3d,C4=2ε0k4Ad Given system of C1,C2,C3 and C4 can be simplified as ∴1CAB=1C1+C2+C3+1C4 Suppose, CAB=kε0Ad1kε0Ad=123ε0Adk1+k2+k3+12ε0Adk4⇒1k=32k1+k2+k3+12k4∴2k=3k1+k2+k3+1k4
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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