Divergence and Curl of Electric Field
Divergence and Curl of Electric Field

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EEE340Lecture 071 2-12 Helmholtz’s Theorem Helmholtz’s Theorem: A vector field (vector point function) is determined to within an additive constant if both its divergence and its curl are specified everywhere.

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EEE340Lecture 072 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. Coulomb’s Law (1785 by Charles Augustive de Coulomb) b. Gauss’s Law Throughout this chapter we will assume that the electric field is in a vacuum or free space.

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EEE340Lecture 073 The force F between two point charges q 1 and q 2 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)

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EEE340Lecture 074 The following table is related to chapter 1:

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EEE340Lecture 075 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 (3.4) (3.5) (3-2) (3.7) (3.8)

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EEE340Lecture 076 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

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EEE340Lecture 077 3-3 Coulomb’s Law Coulomb’s 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)

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EEE340Lecture 078 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

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EEE340Lecture 079 You’ll 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.

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