Checkered Capacitor

A parallel plate capacitor, with plates A\textbf{A} and B\textbf{B} of equal dimensions t×Lt \times L at a distance of LL, is filled with square tiles of dielectric to make a chess-board-like capacitor, as shown in the picture above.

Dielectric constant of the dark tile is σ1,\sigma_1, that of the light tile is σ2,\sigma_2, and ϵ0\epsilon_0 is the permittivity of free space. All the dielectric tiles are square cuboids of thickness tt.

Find the capacitance of this capacitor.

Details and Assumptions:

  • Assume that the electric field varies between the plates like an ideal parallel plate capacitor.

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