[UPSC CSE 2026] Electrostatics and Magnetostatics (Q6, 20M)

The space between the plates of a parallel-plate capacitor is filled with a dielectric material whose susceptibility varies linearly from 0 at the bottom plate (x = 0) to 1 at the top plate (x = d). The capacitor is connected to a battery of voltage V. Calculate all the bound charges and check that the total charge is zero. Assume that the battery is connected in such a way that the electric field points along the x-direction while the free charge density, \sigma_f, is positive at the bottom plate and negative at the top plate.