EQS Solution in 2-D Trough

                 The explicit form of the Laplace equation in two-dimensional Cartesian
                 coordinates is
                                             

                 By using the separation-of-variable technique, (x, y) can be decomposed
                 as , then the Laplace equation can be reduced to

                                             

                   where k is a constant independent of x and y. Possible solutions of

                        are , ,  , , and possible solutions of Y(y) are ,
                   , , or .
                   Fig shows a rectilinear trough with side and bottom walls grounded,
                   and the top wall is connected to a voltage source, . The potential
                   distribution satisfying the boundary conditions that at and
                   is  . The boundary condition that  at
                   renders , with n an integer. The potential can thus be
                   expressed as

                                    
                  The boundary condition at y = b requires that

                                       
                  By utilizing the orthogonality property of the functions, first
                  multiply the above equation by , and integrate over   to have

                  Potential Distribution

                            ,

                             Electric Field Distribution     

                                  

                             Surface Charge Distribution