Exercise 7.2: Numerical solution inside a rectangular region

1. Determine the potential $V(x,y)$ inside a square region with linear dimension $L=10cm$. The boundary of the square is at a fixed potential $V=10V$. Before the computation, guess the exact form of the potential and set the values in the interior within 5the area of each cell to be 1cm$^2$. How many iteration do you need to achieve a 1
2. Consider the same situation, but with $V(5,5)=4$. Describe the evolution of the solution with the iterations. Does the potential distribution evolve o the correct solution? Are the final results independent of your initial guess? What is the effect of a poor initial guess? Note that if you pick a constant values as the intial guess, the result does not evolve toward a solution.

3. Set each side of the rectangle at a different potential 5,10,5,10, repectively. Do a contour plot and sketch the equipotential surfaces. What happens if the potential is 10V on three sides and 0 on the fourth? Start with a reasonable guess for the initial values of the potential of the interior points, and iterate until 1
4. Repeat the previous item with a twice the number of cells, and compare the results.

5. Modify your programs such the potential at each site is calculated sequentially rather than at the same time. How do your reults differ?