The E-field and surface charge density are

1

• The E-field and surface charge density are
• And

(4.15)
2

• Example P.4-23 Determine the potential and
  E-field between two wedge-shaped semi-infinite
  plates, 0ltjlta.
• Solution The voltage is a function of j only,
  I.e. V(j). The Laplace equation reduces to
• The boundary conditions are
• Therefore
• and

y
Vo
a
x
3

• 4-3 Uniqueness of Electrostatic Solution
• A solution to the Laplaces (or Poissons)
  equation that satisfies the given boundary
  conditions is the (only) unique solution,
  regardless what method was employed.
• Proof Textbook pp. 158-159.
• 4-4 Method of Image

4
y

• Observations
• At any point on the ground
• As R
• As
• V is an even function w.s.t. the y-axis, i.e.,
• The E-field is ground.

Q (0,d,0)
x
GND
5

• On the other hand,
• Conclusions
• This problem is equivalent to a configuration of
  an image charge of Q , and then remove the GND.
• The equivalence is only for the upper half plane.
• The lower half plane has zero V and zero . In
  other words, the image solution in the lower half
  plane must be dropped.

6

• Example 4-3 Image method for a conducting corner
• Solution
• where

d1
Q
-Q
d2
-Q
Q
7
4-5 Boundary Value Problems in Cartesian
Coordinates

• In the source free region, Poissons equation
  reduces
• to the Laplace equation
• A standard way to solve the PDE is the method of
• separation of variables, by Fourier

(4-81)
8
Example 4-7 Two-dimensional Laplaces Equations

• with boundary conditions
• Separation V(x, y) X(x) Y(y).
• XY XY 0

y
x
b
0
9
Let us assume
From 1).
10
Boundary conditions
11
Boundary conditions
Hence,
or
12
Let
2. Find unknown constant Cn. 1.)
Superposition (Linear Space)