Complex Analysis - Maths KU

In Section 4.5, for purposes that will
be clear there, solution (11) will be
denoted as φ (x, y).
φ = k 0
y
–1 1
φ = k 1
Figure 3.11 The equipotential curves
and lines of force for Example 2
x
☞
3.4 Applications 169
reasonable to try to seek a solution of (9) of the form φ(x). With this latter
assumption, Laplace’s partial differential equation (9) becomes the ordinary
differential equation d 2 φ/dx 2 = 0. Integrating twice gives the general solution
φ(x) =ax+b. The boundary conditions enable us to solve for the coefficients a
and b. In particular, from φ(−1) = k0 and φ(1) = k1 we must have a(−1)+b =
k0 and a(1) + b = k1, respectively. Adding the two simultaneous equations
gives 2b = k0 + k1, whereas subtracting the first equation from the second
yields 2a = k1 − k0. These two results give us a and b:
b = k1 + k0
2
and a = k1 − k0
.
2
Therefore, we have the following solution of the given Dirichlet problem
φ(x) = k1 − k0
2
x + k1 + k0
. (11)
2
The problem in Example 2 can be interpreted as the determination the
electrostatic potential φ between two infinitely long parallel conducting plates
that are held at constant potentials. Since it satisfies Laplace’s equation in
D, φ is a harmonic function. Hence a harmonic conjugate ψ can be found
as follows. Because φ and ψ must satisfy the Cauchy-Riemann equations, we
have:
∂ψ
∂y
∂φ
=
∂x = k1 − k0
2
and
∂ψ
∂x
= −∂φ
∂y =0.
The second equation indicates that ψ is a function of y alone, and so integrating
the first equation with respect to y we obtain:
ψ(y) = k1 − k0
y,
2
where, for convenience, we have taken the constant of integration to be 0.
From (8), a complex potential function for the Dirichlet problem in Example
2 is then Ω(z) =φ(x)+iψ(y), or
Ω(z) = k1 − k0
x +
2
k1 + k0
2
+ i k1 − k0
y.
2
The level curves of φ or equipotential curves ∗ are the vertical lines x = c1
shown in color in Figure 3.11, and the level curves of ψ or the lines of force
are the horizontal line segments y = c2 shown in black. The figure clearly
shows that the two families of level curves are orthogonal.
∗The level curves of φ are φ(x) =C1 or 1
2 (k1 − k0) x + 1
2 (k1 + k0) =C1. Solving for
x gives x = � C1 − 1
2 k1
�� 1
− k0 2 (k1 − k0). Set the constant on the right side of the last
equation equal to c1.