A parallel plate capacitor consists of two flat plates of area A separated by a distance 3d . A dielectric slab of dielectric constant K = 3 is inserted between the plates. The slab has a stair-like profile with three steps, each of area A 3 . The thicknesses of the dielectric in the three sections are d , 2d , and 3d respectively, with the remaining space in each section filled with air. The equivalent capacitance o

Step-by-step solution

The arrangement can be modeled as three parallel branches, each having a cross-sectional area of A 3 . In each branch, the dielectric and the air gap act as two capacitors in series. For the first branch, the dielectric thickness is d and the air gap is 3d - d = 2d . The equivalent capacitance of this branch is: C₁ = ₀ ( A 3 ) d K + 2d 1 = ₀ A 3 ( d 3 + 2d ) = ₀ A 7d For the second branch, the dielectric thickness is 2d and the air gap is 3d - 2d = d . The equivalent capacitance is: C₂ = ₀ ( A 3 ) 2d K + d 1 = ₀ A