Exercise 2.   Find the electrostatic potential    in the upper half-plane  ,  

that satisfies the boundary values,  (shown in Figure 11.40)  

                      

Solution 2.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Apply Theorem 11.2 to construct a Dirichlet solution in the upper half-plane

use formula (11-5)      with   ,  

                    .  

Now we substitute      and      to obtain  

                       

Therefore,   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Enter the boundary values and construct the Dirichlet form of the solution.

We are really done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function   

                     where      for   .  

We are really really done.   

                     A contour graph of the function   

                     where      for   .  

                    A graph of the function   ,

                       

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell