1. 3.3 Separation of Variables 3.4 Multipole Expansion
Chapter 3 Special Techniques for Calculating Potential
3.1 Laplace’s Equation
3.2 The Method of Images
3.3 Separation of Variables
3.4 Multipole Expansion

2. 3.1.2 Laplace’s Equation in One Dimension
Introduction
Laplace’s Equation in One Dimension
Laplace’s Equation in Two Dimensions
Laplace’s Equation in Three Dimensions
Boundary Conditions and Uniqueness
Theorems
Conducts and the Second Uniqueness
Theorem

3. 3.1.1 Introduction
The primary task of electrostatics is to study the interaction
(force) of a given stationary charges.
since
this integrals can be difficult (unless there is symmetry)
we usually calculate
This integral is often too tough to handle analytically.

4. 3.1.1 In differential form Eq.(2.21): Poisson’s eq.
to solve a differential eq. we need boundary conditions.
In case of ρ = 0, Poisson’s eq. reduces to
Laplace’s eq or
The solutions of Laplace’s eq are called harmonic function.

5. 3.1.2 Laplace’s Equation in One Dimension
m, b are to be determined by B.C.s
e.q.,
V=4
V=0 x 1 5
1. V(x) is the average of V(x + R) and V(x - R), for any R:
2. Laplace’s equation tolerates no local maxima or minima.

6. 3.1.2 (2) Method of relaxation:
A numerical method to solve Laplace equation.
Starting V at the boundary and guess V on a grid of interior points.
Reassign each point with the average of its nearest neighbors.
Repeat this process till they converge.

7. 3.1.2 (3)
The example of relaxation

8. 3.1.3 Laplace’s Equation in Two Dimensions
A partial differential eq. :
There is no general solution. The solution will be given in 3.3.
We discuss certain general properties for now.
The value of V at a point (x, y) is the average of those around the point.
2. V has no local maxima or minima; all extreme occur at the boundaries.
3. The method of relaxation can be applied.

9. 3.1.4 Laplace’s Equation in Three Dimensions
1. The value of V at point P is the average value of V
over a spherical surface of radius R centered at P:
The same for a collection of q by the superposition principle.
2. As a consequence, V can have no local maxima or minima;
the extreme values of V must occur at the boundaries.
3. The method of relaxation can be applied.

10. 3.1.5 Boundary Conditions and Uniqueness Theorems
in 1D,
one end
the other end Va Vb
V is uniquely determined by its value at the boundary.
First uniqueness theorem :
the solution to Laplace’s equation in some region is uniquely
determined, if the value of V is specified on all their surfaces;
the outer boundary could be at infinity, where V is ordinarily taken
to be zero.

11. 3.1.5 Proof: Suppose V1 , V2 are solutions at boundary at boundary.
V3 = everywhere
hence V1 = V2 everywhere

12. 3.1.5 The first uniqueness theorem applies to regions with charge.
Proof.
at boundary.
Corollary : The potential in some region is uniquely determined if
(a) the charge density throughout the region, and
(b) the value of V on all boundaries, are specified.

13. 3.1.6 Conductors and the Second Uniqueness Theorem
In a region containing conductors and filled with a specified charge density ρ,
the electric field is uniquely determined if the total charge on each conductor is given.
(The region as a whole can be bounded by another conductor, or else unbounded.)
Proof:
Suppose both and are satisfied.
and

14. 3.1.6
define
for V3 is a constant over each conducting surface

15. 3.2 The Method of Images
3.2.1 The Classical Image Problem
3.2.2 The Induced Surface Charge
3.2.3 Force and Energy
3.2.4 Other Image Problems

16. 3.2.1 The Classical Image Problem
B.C.
What is V (z>0) ?
The first uniqueness theorem guarantees that
there is only one solution.
If we can get one any means, that is the only answer.

17. 3.2.1 Trick : Only care z > 0 z < 0 is not of concern
V(z=0) = const = 0
in original problem
for

18. 3.2.2 The Induced Surface Charge
Eq.(2.49)

19. 3.2.2
total induced surface charge
Q=-q

20. 3.2.3 Force and Energy The charge q is attracted toward the plane.
The force of attraction is
With 2 point charges and no conducting plane, the energy is
Eq.(2.36)

21. 3.2.3 (2)
For point charge q and the conducting plane at z = 0 the energy is half of the energy given at above, because the field exist
only at z ≥ 0 ,and is zero at z < 0 ; that is or

22. 3.2.4 Other Image Problems
Stationary distribution of charge

23. 3.2.4 (2)
Conducting sphere of radius R
Image charge at

24. 3.2.4 (3)
Force
[Note : how about conducting circular cylinder?]

25. 3.3 Separation of Variables
3.3.0 Fourier series and Fourier transform
3.3.1 Cartesian Coordinate
3.3.2 Spherical Coordinate

26. 3.3.0 Fourier series and Fourier transform
Basic set of unit vectors in a certain coordinate can express
any vector uniquely in the space represented by the coordinate.
e.g.
in 3D. Cartesian Coordinate.
are unique because
are orthogonal. = 1
Completeness: a set of function
is complete. if
for any function
Orthogonal: a set of functions is orthogonal if
=const
for

27. 3.3.0 (2) A complete and orthogonal set of functions forms
a basic set of functions.
e.g.
odd
even
sin(nx) is a basic set of functions for any odd function.
cos(nx) is a basic set of functions for any even function.
sin(nx) and cos(nx) are a basic set of functions for any functions.

28. 3.3.0 (3) , for any odd ; even odd even [sinkx] [coskx]
Fourier transform and Fourier series

29. 3.3.0 (4)
Proof

30. 3.3.0 (5)
B0 2p for k = 0
Bk p for k = 1, 2, …

31. 3.3.1 Cartesian Coordinate
Use the method of separation of variables to solve the Laplace’s eq.
Example 3
Find the potential inside this “slot”?
Laplace’s eq.
set

32. 3.3.1 (2)
f and g are constant
set so

33. 3.3.1 (3) B.C. (iv) B.C. (i) B.C. (ii) The principle of superposition
B.C. (iii)
A fourier series for odd function

34. 3.3.1 (4)
For

35. 3.4 Multipole Expansion
3.4.1 Approximate Potentials at Large distances
3.4.2 The Monopole and Dipole Terms
3.4.3 Origin of Coordinates
in Multipole Expansions
3.4.4 The Electric Field of a Dipole

36. 3.4.1 Approximate Potentials at Large distances
Example 3.10 A dipole (Fig 3.27). Find the approximate potential
at points far from the dipole.

37. 3.4.1 Example 3.10 For an arbitrary localized charge distribution.
Find a systematic expansion of the potential?

38. 3.4.1(2)
Monopole term
Dipole term
Quadrupole term
Multipole expansion

39. 3.4.2 The Monopole and Dipole Terms
dominates if r >> 1
dipole
dipole moment (vector)
A physical dipole is consist of a pair of equal and
opposite charge,

40. 3.4.3 Origin of Coordinates in Multipole Expansionsif

41. 3.4.4 The Electric Field of a Dipole
A pure dipole