Lecture Engineering electromagnetics - Vector analysis presents the following content: Scalars and vectors, the rectangular coordinate system, the Dot product and the cross product, the circular cylindrical coordinate system, the spherical coordinate system.
Trang 1Engineering Electromagnetics
Nguy ễ n Công Ph ươ ng
Trang 2I Introduction
II Vector Analysis
III Coulomb’s Law & Electric Field Intensity
IV Electric Flux Density, Gauss’ Law & Divergence
V Energy & Potential
VI Current & Conductors
VII Dielectrics & Capacitance
VIII.Poisson’s & Laplace’s Equations
IX The Steady Magnetic Field
X Magnetic Forces & Inductance
XI Time – Varying Fields & Maxwell’s Equations
XII The Uniform Plane Wave
XIII.Plane Wave Reflection & Dispersion
Trang 3Introduction (1)
• The study of charges (at rest or in motion)
• Fundamental to electrical engineering
Trang 4Introduction (2)
Magnetostatics
0
I t
∂ ≠
∂
Electromagnetic Waves
Trang 6I Introduction
II Vector Analysis
III Coulomb’s Law & Electric Field Intensity
IV Electric Flux Density, Gauss’ Law & Divergence
V Energy & Potential
VI Current & Conductors
VII Dielectrics & Capacitance
VIII.Poisson’s & Laplace’s Equations
IX The Steady Magnetic Field
X Magnetic Forces & Inductance
XI Time – Varying Fields & Maxwell’s Equations
XII The Uniform Plane Wave
XIII.Plane Wave Reflection & Dispersion
Trang 7Vector Analysis
1 Scalars & Vectors
2 The Rectangular Coordinate System
3 The Dot Product & The Cross Product
4 The Circular Cylindrical Coordinate System
5 The Spherical Coordinate System
Trang 8Scalars & Vectors
• Scalar: refers to a quantity whose value may be
represented by a single (positive/negative) real number
• Ex.: distance, time, temperature, mass, …
• Scalars are in italic type, e.g t, m, E,…
• Vector: refers to a quantity whose value may be
represented by a magnitude and a direction in space (2D,
3D, nD)
• Ex.: force, velocity, acceleration, …
• Vectors are in bold type, e.g A
• A may be written as A
Trang 9Vector Analysis
1 Scalars & Vectors
2 The Rectangular Coordinate System
3 The Dot Product & The Cross Product
4 The Circular Cylindrical Coordinate System
5 The Spherical Coordinate System
Trang 10The Rectangular Coordinate System (1)
x
y z
0
Trang 11The Rectangular Coordinate System (2)
Trang 12The Rectangular Coordinate System (3)
Trang 13The Rectangular Coordinate System (4)
Trang 14The Rectangular Coordinate System (5)
Given a vector V = 5ax – 2ay + 4az, find:
a) Its components?
b) Its magnitude?
c) Its unit vector ?
Ex.
Trang 15Vector Analysis
1 Scalars & Vectors
2 The Rectangular Coordinate System
3 The Dot Product & The Cross Product
4 The Circular Cylindrical Coordinate System
5 The Spherical Coordinate System
Trang 16The Dot Product (1)
Trang 17The Dot Product (2)
B a
θ Ba
B·a
B a
θ Ba
(B·a)a
The scalar component
of B in the direction of
the unit vector a
The vector component
of B in the direction of the unit vector a
Trang 18The Dot Product (3)
Consider the vector field G = zax – 2xay + 3yaz and the point Q(4, 3, 2) Find:
a) G at Q ?
b) The scalar component of G at Q in the direction of aN = ⅓(ax + 2ay – 2az) ?
c) The vector component of G at Q in the direction of aN ?
d) The angle between G(rQ) & aN ?
Ex.
a) ( G r Q ) = 2 a x − × 2 4 a y + × 3 3 a z = 2 a x − 8 a y + 9 a z
1 b) (2 8 9 ) ( 2 2 )
3 1
(2 1 8 2 9 2) 10.67 3
Trang 19The Dot Product (4)
Consider the vector field G = zax – 2xay + 3yaz and the point Q(4, 3, 2) Find:
a) G at Q ?
b) The scalar component of G at Q in the direction of aN = ⅓(ax + 2ay – 2az) ?
c) The vector component of G at Q in the direction of aN ?
d) The angle between G(rQ) & aN ?
Ex.
1
3 3.55 7.11 7.11
Trang 20The Dot Product (5)
Consider the vector field G = zax – 2xay + 3yaz and the point Q(4, 3, 2) Find:
a) G at Q ?
b) The scalar component of G at Q in the direction of aN = ⅓(ax + 2ay – 2az) ?
c) The vector component of G at Q in the direction of aN ?
d) The angle between G(rQ) & aN ?
Trang 21The Cross Product (1)
• A B = a N |A||B|sinθ AB
– a N : normal (unit) vector
• B A = – (A B)
A B
Trang 22The Cross Product (2)
Given A = ax – 2ay + 3az and B = –4ax + 5ay – 6az Find their cross product ?
Trang 23The Cross Product (3)
Given A = ax – 2ay + 3az and B = –4ax + 5ay – 6az Find the angle between A & B?
Trang 24The Cross Product (4)
Given A = ax – 2ay + 3az and B = –4ax + 5ay – 6az Find the angle between A & B?
Trang 25The Cross Product (5)
Given A = ax – 2ay + 3az, B = –4ax + 5ay – 6az, and C = ax – ay + az Find:
Trang 26The Rectangular Coordinate System (6)
Trang 27Vector Analysis
1 Scalars & Vectors
2 The Rectangular Coordinate System
3 The Dot Product & The Cross Product
4 The Circular Cylindrical Coordinate System
5 The Spherical Coordinate System
Trang 28The Circular Cylindrical Coordinate System (1)
Trang 29The Circular Cylindrical Coordinate System (2)
z+dz
z
dρ dz
ρdφ
dS = ρdρdφa z
dS = ρdφdza ρ
dS= dρdza φ
Trang 30The Circular Cylindrical Coordinate System (3)
Trang 31Vector Analysis
1 Scalars & Vectors
2 The Rectangular Coordinate System
3 The Dot Product & The Cross Product
4 The Circular Cylindrical Coordinate System
5 The Spherical Coordinate System
Trang 32The Spherical Coordinate System (1)
Trang 33The Spherical Coordinate System (1)
Trang 34The Spherical Coordinate System (1)
x
y z
φ
0
φ
Trang 35The Spherical Coordinate System (2)
x
y z
φ θ
r
Trang 36The Spherical Coordinate System (3)
dS = rsinθdrdφa θ
dS = r 2 sinθdθdφa r
dS = rdrdθa φ
Trang 37The Spherical Coordinate System (4)
x
y z
Trang 38RECTANGULAR CYLINDRICAL SPHERICAL