Exercise 6.   The complex potential    determines an electrostatic field that is referred to as a dipole.

6 (b).   Show that the lines of flux of a dipole are circles that pass through the origin,

as shown in Figure 11.104.                     Figure 11.104.

Solution 6 (b) .

Consider Example 11.30 where a source is located at   and sink is located at  .  The streamlines were shown to be circles that satisfy the equation  

            ,  

If the strength is changed to .  The velocity potential and stream function are  

            ,   and  

            ,  

respectively.  Solving for the streamline  ,  we start with   

              

Then get the equation   which can be written as

            .

            
            
            

            

              

which is the equation of a circle with center at    that passes through the points  .

If we let in the last equation, and observe that    and    then we obtain

              

              
        
which is the equation of a circle with center at    and radius .

We are done.   

Aside.  We can let Mathematica double check our work.

                     Graphs of some circles   .  

We are really done.   

Aside.  We can let Mathematica graph some of the equipotential curves and flux lines.

The velocity potential    is   

The stream function     is   

                    Some equipotential curves   .    

                    A density plot for the  equipotentials   .    

                    Some flux lines   .  

This solution is complements of the authors.

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