11.6 Two-Dimensional Electrostatics

    A two-dimensional electrostatic field is produced by a system of charged wires, plates, and cylindrical conductors that are perpendicular to the z plane. The wires, plates, and cylinders are assumed to be so long that the effects at the ends can be neglected, as mentioned in Section 11.4.  This assumption results in an electric field    that can be interpreted as the force acting on a unit positive charge placed at the point .  In the study of electrostatics the vector field    is shown to be conservative and is derivable from a function  ,  called the electrostatic potential, expressed as  

            .

    If we make the additional assumption that there are no charges within the domain D,  then Gauss' Law for electrostatic fields implies that the line integral of the outward normal component of     taken around any small rectangle lying inside D is identically zero.  A heuristic argument similar to the one for steady state temperatures with    replaced by     will show that the value of the line integral is  
    
            .  

This quantity is zero, so we conclude that    is a harmonic function.  If we let    be the harmonic conjugate, then  

              

is the complex potential (not to be confused with the electrostatic potential).

The curves    are called the equipotential curves, and the curves    are called the lines of flux.  If a small test charge is allowed to move under the influence of the field   ,  then it will travel along a line of flux.  Boundary value problems for the potential function are mathematically the same as those for steady state heat flow, and they are realizations of the Dirichlet problem where the harmonic function is .

Example 11.18.  Consider two parallel conducting planes that pass perpendicular to the z plane through the lines  ,  which are kept at the potentials   and , respectively.  Then according to the result of Example 11.1, (see Section 11.1), the electrical potential is

            .  

Explore Solution 11.18.

Example 11.19.  Find the electrical potential    in the region between two infinite coaxial cylinders  ,  which are kept at potentials  ,  respectively.  

Solution.  The transformation  

              

maps the annular region between the circles    onto the infinite strip    in the w plane, as shown in Figure 11.36.  The potential    in the infinite strip has the boundary values  

               and      for all  v.

If we use the result of Example 11.18, the electrical potential    is  

            .  

Because  ,  we can use this equation to conclude that the potential    is  

            .  

The equipotentials    are concentric circles centered on the origin, and the lines of flux are portions of rays emanating from the origin.  If  ,  then the situation is as illustrated in Figure 11.36.

Figure 11.36  The electrical field in a coaxial cylinder, where .

Explore Solution 11.19.

Example 11.20.   Find the electrical potential    produced by two charged half-planes that are perpendicular to the z plane and pass through the rays    where the planes are kept at the fixed potentials  

              

Solution.  The result of Example 10.13, (see Section 10.4), shows that the function  

              

is a conformal mapping of the z plane slit along the two rays    onto the vertical strip  . The new problem is to find the potential    that satisfies the boundary values  

              

From Example 11.1, (see Section 11.1),

            .  

As in the discussion of Example 11.17, (see Section 11.5),the solution in the z plane is

            .  

Several equipotential curves are shown in Figure 11.37.

Figure 11.37  The electric field produced by two charged half-planes
                    that are perpendicular to the complex plane.

Explore Solution 11.20.

Example 11.21.  Find the electrical potential    in the disk    that satisfies the boundary values

              

Solution.  The mapping    is a one-to-one conformal mapping of D onto the upper half-plane    with the property that is mapped onto the negative u axis and is mapped onto the positive u axis.  The potential in the upper half-plane that satisfies the new boundary values

              

is given by  

(11-29)            .  

A straightforward calculation shows that  

            

We substitute the real and imaginary parts,    and    from this equation, into Equation (11-29) to obtain the desired solution:   

             .  

The level curve    in the upper half-plane is a ray emanating from the origin, and the preimage    in the unit disk is an arc of a circle that passes through the points  .  Several level curves are illustrated in Figure 11.38.

            Figure 11.38  The potentials    and  . 

Explore Solution 11.21.

Extra Example 1.  Find the electrical potential    in the crescent-shaped region that lies inside the disk   and outside the circle   that satisfies the following boundary values shown in Figure 11.41:

              

Solution.  The result of Example 10.7, (see Section 10.2), shows that the function  

              

is a conformal mapping of the crescent-shaped region that lies inside the disk   and outside the circle   onto the horizontal strip  .  

Figure 11.41  The electrical potential inside   and outside .  

Revisit and Explore Solution 10.7.

    The new problem in the w plane is to find the potential    that satisfies the boundary values  

              
            
In the w plane the solution is

            

A straightforward calculation shows that  

              

We substitute the imaginary part,    from this equation, into    to obtain the desired solution:   

               

Explore Extra Solution 1.

Extra Example 2.  Find the electrical potential    in the semi-infinite strip that has the boundary values shown in Figure 11.42:

              

Solution.  The result of Example 10.13, (see Section 10.4), shows that the function  

               

is a conformal mapping of the semi-infinite strip    onto the upper half-plane  .

    The new problem in the w plane is to find the electrical potential    that satisfies the boundary values  

            

This is a three-value Dirichlet problem in the upper half-plane defined by  .  For the w plane, the solution in Equation (11-5) becomes  

            

Here we have    and  ,  which we substitute into the above equation for    to obtain  

                

Now make the substitutions    and    to get the solultion    in the z plane  

            

            

Figure 11.42  The electrical potential in the semi-infinite strip .  

Explore Extra Solution 2.

Exercises for Section 11.6.  Two-Dimensional Electrostatics