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

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 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 (A) k = k 1 + k2 + k3 + 3k4 (B) k = (2/3) (k1 + k2 + k3) + 2k4 (C) (2/k) = (3/k1 + k2 + k3) + (1/k4) (D) (1/k) = (1/k1 ) + (1/k2) + (1/k3) + (3/2 k4)

Tardigrade Admin · Accepted Answer

(1/C) = (1/C1) + (1/C2) (d/kA ∈0 ) = (3d/2(k1 + k2+k3)A ∈0) + (d/2k4A ∈0) (d/kA ∈0) = (d/A ∈0) [(3/3(k1 + k2 +k3)) + (1/2k4)] (2/k) = (3/k1 + k2 +k3) = (1/k4 )

$k = k_{1} + k_{2} + k_{3} + 3 k_{4}$

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

$\frac{2}{k} = \frac{3}{k _{1} + k _{2} + k _{3}} + \frac{1}{k _{4}}$

$\frac{1}{k} = \frac{1}{k _{1}} + \frac{1}{k _{2}} + \frac{1}{k _{3}} + \frac{3}{2 k _{4}}$

k=k1​+k2​+k3​+3k4​

k=32​(k1​+k2​+k3​)+2k4​

k2​=k1​+k2​+k3​3​+k4​1​

k1​=k1​1​+k2​1​+k3​1​+2k4​3​

C1​=C1​1​+C2​1​ kA∈0​d​=2(k1​+k2​+k3​)A∈0​3d​+2k4​A∈0​d​ kA∈0​d​=A∈0​d​[3(k1​+k2​+k3​)3​+2k4​1​] k2​=k1​+k2​+k3​3​=k4​1​

$k = k_{1} + k_{2} + k_{3} + 3 k_{4}$

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

$\frac{2}{k} = \frac{3}{k _{1} + k _{2} + k _{3}} + \frac{1}{k _{4}}$

$\frac{1}{k} = \frac{1}{k _{1}} + \frac{1}{k _{2}} + \frac{1}{k _{3}} + \frac{3}{2 k _{4}}$