$1)$Write a Python program that finds the electrostatic potential $$ as function

of $x$ and $y$ in $.$ Solve the algebraic system using a direct solver, and choose

a grid spacing $h$ that gives a reasonably well$-$resolved solution, while not

requiring overly much computational time. Discuss your choice of $h.$

$2)$Display the solution $(x,y)$ as a contour plot $($use contourf function in

matplotlib and meshgrid function in numpy$)$ for the potential $$ and a vector

feld distribution $($use quiver and streamplot functions in matplotlib$)$ for

the electrostatic field $E($for computing the electrostatic field you can use the

function gradient in numpy$).$

$3)$Calculate the electrostatic energy $E$ stored per unit length $($i$.$e$.\frac{J}{m})$ for this

problem, defined by:

$E={\displaystyle \underset{}{\overset{\phantom{}}{}}}\frac{{}_0}{2}{E}^2(x,y)d$

$4)$Deduce the value of the capacitance $C(\frac{F}{m})$ of the capacitor. Compare with

the classical formula of the planar capacitor for several geometries of the

device $($i$.$e$.$ vary the dimensions of the armatures, their spacing, as well as

the dimensions of the box$)$ and plot the evolution of capacity as a function of

variation in the parameter$($s$)$ under consideration.

We now consider the presence of a dielectric material with relative dielectric

permittivity ${}_r$ between the two electrodes $($Figure $2).$

$5)$From the calculation of electrostatic energy, calculate and plot the evolution

of capacitance $C,(\frac{F}{m})$ for ${}_{r}in[1,200].$

Figure $2$: General study domain of the planar capacitor with a dieiectric material