Exercise 6.   Find the electrostatic potential    in the infinite strip   ,  

that satisfies the following boundary values,  (shown in Figure 11.44)  

                      

Hint.  Use  .

Solution 6.

See text and/or instructor's solution manual.

Answer.   Map the infinite strip onto the right half plane slit along the ray    with the mapping  ,  

then construct   .   

Solution.   The transformation    maps the infinite strip    

onto the right half plane slit along the ray  .     

                      The mapping   .  

Now construct the intermediate solution    in the w-plane that has the boundary values

                      
        
Notice that the intermediate solution      will agree with these boundary values (and also for ).  

Therefore the solution in the infinite strip      is   

                    ,  

                    .  

Observation.  There is an equipotential   for   that is the preimage of the segment    where we have  .

We are done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function     

                     where      for   .  

We are really really done.   

                     A graph of the function   ,   

                      

This solution is complements of the authors.

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