Question #215604

A parallel-plate capacitor of a plate area A and plate separation d. The left half

of the gap is filled with a material with dielectric constant κ1; the top of the right half is filled with

material of dielectric constant κ2; the bottom of the right half is filled material of dielectric constant

κ3. What is the capacitance C?


1
Expert's answer
2021-07-12T16:22:18-0400

Gives

Dielectric material=k1,k2,k3k_1,k_2,k_3

Let us assume d'=2d


C1=ϵ0(A2)k12d=Aϵ0k14dC_1=\frac{\epsilon_0( \frac{A}{2} )k_1}{2d'}=\frac{A\epsilon_0 k_1}{4d'}

C2=ϵ0(A2)k2d=Aϵ0k22dC_2=\frac{\epsilon_0( \frac{A}{2} )k_2}{d'}=\frac{A\epsilon_0 k_2}{2d'}

C3=ϵ0(A)k32dC_3=\frac{\epsilon_0( {A}{} )k_3}{2d'}

C=C1+C2C3C1+C3C=C_1+\frac{C_2C_3}{C_1+C_3}

C=Aϵ04d[k1+2k2k3k2+k3]C=\frac{A\epsilon_0}{4d'}[k_1+\frac{2k_2k_3}{k_2+k_3}]

C=Aϵ02d[k12+k2k3k2+k3]C=\frac{A\epsilon_0}{2d'}[\frac{k_1}{2}+\frac{k_2k_3}{k_2+k_3}]

C=Aϵ0d[k12+k2k3k2+k3]C=\frac{A\epsilon_0}{d}[\frac{k_1}{2}+\frac{k_2k_3}{k_2+k_3}]



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