Problem $5$

The space between two parallel conducting plates each having an area $S$ is filled with an inhomogeneous ohmic medium whose conductivity varies linearly from ${}_1$ at $y=0$ to ${}_2$ at $y=d.$ A DC voltage $V_o$ is applied across the plates as shown in figure. Determine

a$)$ The total resistance between the plates

b$)$ The surface charge density on each plate

c$)$ The volume charge density and the total amount of charge between the plates