Checkered Capacitor

A parallel plate capacitor, with plates \(\textbf{A}\) and \(\textbf{B}\) of equal dimensions \(t \times L\) at a distance of \(L\), 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 \(\sigma_1,\) that of the light tile is \(\sigma_2,\) and \(\epsilon_0\) is the permittivity of free space. All the dielectric tiles are square cuboids of thickness \(t\).

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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