2-12 Helmholtz

1
2-12 Helmholtzs Theorem

• Helmholtzs Theorem A vector field (vector
  point function) is determined to within an
  additive constant if both its divergence and its
  curl are specified everywhere.

2
Chapter 3 Static Electric (Electrostatic)
Fields

• 3-1 Introduction
• An electrostatic field is produced by a static
  charge distribution. It is time-invariant.
• There are two fundamental laws governing
  electrostatic fields
• a. Coulombs Law (1785 by Charles Augustive de
  Coulomb)
• b. Gausss Law
• Throughout this chapter we will assume that the
  electric field is in a vacuum or free space.

3

• The force F between two point charges q1 and q2
  is
• and it acts along the line joining them. Here, R
  is the
• distance between charges k is the
  proportionality constant.
• ?o is known as the permittivity of free space
  (F/m)
• (in MKS)
• (in CGS)
• or
• In this book we use only MKS

(3.1)
4
The following table is related to chapter 1
5
3-2 Fundamental Postulates of Electrostatics in
Free Space

• Electric field intensity (E-field)
• Two postulations of electrostatics in free-space
• Divergence
• Curl
• The corresponding integral form
• 
  
• 
  

(3-2)
(3.4)
(3.5)
(3.7)
(3.8)
6

• or
• or

Zero scalar line integral implies conservation of
energy, I.e., the work produced is independent of
the path, but depends only on the starting and
ending points (states). Electrostatic field is
conservative. Eq. (3-7) says that electrostatic
field has a source Q
7
3-3 Coulombs Law

• Coulombs Law states for a point charge at the
  origin
• If the charge is not at the origin, bur at
  , then

(3.12)
(3.13)
(3.15)
(3.14)
8

• Note that is a new vector pointing
  from the source to field point
• The uni-vector
• For the convenience in (2-32)-(2-37),
  (3-61)-(3-63), later sections/chapters, and most
  EM books and Journals, let us use
• As a result, the uni-vector

9

• Youll get used to the notation.
• In case you are not sure whether in a
  equation is a distance vector or a vector in
  spherical coordinates, you are free to ask.