Show all steps and computations. $9\mathrm{.}21$ The Annulus and the Trapezoid The annulus shown below is cut from a planar metal sheet with thickness $\backslash ($ t $\backslash )$ and conductivity $\backslash (\backslash$sigma $\backslash ).$

$($a$)$ Let $\backslash ($ V $\backslash )$ be the voltage between the edge $\backslash ($ C D $\backslash )$ and the edge $\backslash ($ F A $\backslash ).$ Solve Laplace's equation to find the electrostatic potential, current density, and resistance of the annulus.

$($b$)$ Divide the annulus in a sequence of concentric sub$-$annuli, each with width $\backslash ($ d r $\backslash ).$ Show how to combine the resistances of the individual sub$-$annuli to reproduce the resistance computed in part $($a$).$ Use the lines of current density predicted in each case to explain why the two calculations agree.

$($c$)$ Let $\backslash ($ V $\backslash )$ be the voltage between the edge $\backslash ($ A B C $\backslash )$ and the edge $\backslash ($ D E F $\backslash )$ of the original annulus. Repeat all the steps of part $($a$)$ and part $($b$).$

$($d$)$ The trapezoid shown below is cut from a planar metal sheet with thickness $\backslash ($ t $\backslash )$ and conductivity $\backslash (\backslash$sigma $\backslash ).$ Let $\backslash ($ V $\backslash )$ be the voltage between the edge $\backslash ($ A B $\backslash )$ and the edge $\backslash ($ C D $\backslash ).$ Explain why the exact resistance computed by solving Laplace's equation for the entire trapezoid is not the same as the resistance computed by summing the resistances for sub$-$trapezoids like the one indicated by shading in the figure below. Does the summation calculation overestimate or underestimate the exact resistance?