Page 180 - Engineering Electromagnetics, 8th Edition

162 ENGINEERING ELECTROMAGNETICS

2
or conductors produces a field for which ∇ V = 0. All these fields are different,
with different potential values and different spatial rates of change, yet for each
2
of them ∇ V = 0. Because every field (if ρ ν = 0) satisfies Laplace’s equation,
how can we expect to reverse the procedure and use Laplace’s equation to find one
specific field in which we happen to have an interest? Obviously, more information is
required, and we shall find that we must solve Laplace’s equation subject to certain
boundary conditions.
Every physical problem must contain at least one conducting boundary and usu-
ally contains two or more. The potentials on these boundaries are assigned values,
perhaps V 0 , V 1 ,... ,or perhaps numerical values. These definite equipotential sur-
faces will provide the boundary conditions for the type of problem to be solved. In
other types of problems, the boundary conditions take the form of specified values of
E (alternatively, a surface charge density, ρ S )onan enclosing surface, or a mixture
of known values of V and E.
Before using Laplace’s equation or Poisson’s equation in several examples, we
muststatethatifouranswersatisfiesLaplace’sequationandalsosatisfiestheboundary
conditions, then it is the only possible answer. This is a statement of the Uniqueness
Theorem, the proof of which is presented in Appendix D.

D6.5. Calculate numerical values for V and ρ ν at point P in free space if:
4yz π
2
(a) V = ,at P(1, 2, 3); (b) V = 5ρ cos 2φ,at P(ρ = 3,φ = ,
2
x + 1 3
2 cos φ
◦
◦
z = 2); (c) V = ,at P(r = 0.5,θ = 45 , φ = 60 ).
r 2
3
Ans. 12 V, −106.2 pC/m ; −22.5 V, 0; 4 V, 0
6.7 EXAMPLES OF THE SOLUTION
OF LAPLACE’S EQUATION
Several methods have been developed for solving Laplace’s equation. The simplest
method is that of direct integration. We will use this technique to work several exam-
ples involving one-dimensional potential variation in various coordinate systems in
this section.
The method of direct integration is applicable only to problems that are “one-
dimensional,” or in which the potential field is a function of only one of the three
coordinates. Since we are working with only three coordinate systems, it might seem,
then, that there are nine problems to be solved, but a little reflection will show that
a field that varies only with x is fundamentally the same as a field that varies only
with y. Rotating the physical problem a quarter turn is no change. Actually, there are
only five problems to be solved, one in rectangular coordinates, two in cylindrical,
and two in spherical. We will solve them all.
First, let us assume that V is a function only of x and worry later about which
physical problem we are solving when we have a need for boundary conditions.
Laplace’s equation reduces to
2
∂ V = 0
∂x 2