Begin2.DVI

That is

grad φ=

∂φ

∂x

ˆe 1 +∂φ

∂y

ˆe 2 =∂ψ

∂y

ˆe 1 −∂ψ

∂x

ˆe 2. (9 .70)

Differentiate the Cauchy–Riemann equations (9.69) and show

∂^2 φ

∂x^2

= ∂

(^2) ψ
∂y ∂x

and ∂

(^2) φ
∂y^2
=−∂
(^2) ψ
∂x ∂y

. (9 .71)

Addition of these equations produces the Laplace equation

∇^2 φ=

∂^2 φ

∂x^2 +

∂^2 φ

∂y^2 = 0. (9 .72)

Similarly, it can be shown that

∂^2 ψ

∂x^2

= ∂

(^2) φ
∂x ∂y

and ∂

(^2) ψ
∂y^2
= ∂
(^2) φ
∂y ∂x

so that by addition

∇^2 ψ=∂

(^2) ψ
∂x^2
+∂
(^2) ψ
∂y^2
= 0. (9 .73)

Hence, both φand ψare solutions of Laplace’s equation.

Definition: (Harmonic function) Any function which is

a solution of Laplace’s equation ∇^2 ω= 0 and which has

continuous second-order derivatives is called a harmonic

function.

Orthogonality of Equipotential Curves and Field Lines

To show that the equipotential curves φ= c 1 and the field lines ψ = c∗ are

orthogonal, consider the dot product of the vectors normal to these curves at a

common point of intersection. These normal vectors are

grad φ=

∂φ

∂x

ˆe 1 +∂φ

∂y

eˆ 2 and grad ψ=∂ψ

∂x

ˆe 1 +∂ψ

∂y

ˆe 2

and their dot product produces

grad φ·grad ψ=∂φ

∂x

∂ψ

∂x

+∂φ

∂y

∂ψ

∂y

.

With the use of the Cauchy-Riemann equations it can be shown that this dot prod-

uct is zero. Thus the vector grad ψ is perpendicular to the vector grad φand the

equipotential curves and field lines are orthogonal.