Electromagnetics NANENG 331
Fall 2020
Dr Mohamed Farhat
Professor, Senior Member IEEE, OSA
Nanotechnology and Nanoelectronics Engineering Program
Zewail City of Science and Technology
Electromagnetics
NANENG 331- Fall 2020
Dr Mohamed Farhat
Professor, Senior Member IEEE, OSA
Nanotechnology and Nanoelectronics Engineering Program
Zewail City of Science and Technology
Chapter 1
Revision on Vector Analysis & Coordinate Systems
Outline
1-Scalar and Vector Quantities
2- Dot and Cross Products
3- Coordinate Systems
•Cartesian Coordinates
•Cylindrical Coordinates
• Spherical Coordinates
Chapter 1 Vector Analysis
Scalars and Vectors
• Scalar refers to a quantity whose value may be represented by
a single (positive or negative) real number.
• Some examples include distance, temperature, mass, density,
pressure, volume, and time.
Chapter 1 Vector Analysis
Scalars and Vectors
• A vector quantity has both a magnitude and a direction in
space. We especially concerned with two- and
three-dimensional spaces only.
• Displacement, velocity, acceleration, and force are examples of
vectors.
• Scalar notation:
• Vector notation:
arrow)
A or
A (italic or plain)
→
A or A
(bold or plain with
Vector in Cartesian Coordinates
A vector
in Cartesian Coordinates may be represented by three
component vectors, which are Ax, Ay and Az.
OR
The magnitude of A is written as |A|:
6
Unit Vectors
• A unit vector along vector A is;
• A vector with magnitude = 1 (unity)
• Directed along the coordinate axes in the
direction of increasing coordinate values
7
Unit Vectors
Unit vector in the direction of vector A is
8
Example 1: Unit Vector
Specify the unit vector extending from the origin towards the point
Construct the vector extending from origin to point G
Find the magnitude of
9
Vector Algebra
• For addition and subtraction of A and B,
10
Chapter 1 Vector Analysis
Vector Algebra
Two vectors may be added graphically either by:
Drawing both vectors
from a common origin and
completing the
parallelogram
1-
2- By beginning the second vector
from the head of the first and
completing the triangle; either
method is easily extended to three
or more vectors
Chapter 1
Vector Algebra
Commutative Law
Associative Law
Example 2: Vector Algebra
If
Find:
(a) The component of
along
(b) The magnitude of
(c) A unit vector
along
13
Solution to Example 2
(a) The component of
along
is
(b)
14
Solution to Example 2
Hence, the magnitude of
(c)
A unit vector
is:
along
Let
15
Chapter 1 Vector Analysis
Solution to Example 2
So, the unit vector along
is:
16
Position and Distance Vectors
17
Position and Distance Vectors
❑ A point P in Cartesian coordinate
maybe represented as
❑ The position vector
of point P is the
vector from origin O to point P
18
Position and Distance Vectors
If we have two
position vectors,
and
, the third
vector or distance
vector can be
defined as:-
Example 3: Position Vectors
Point P and Q are located at
and
. Calculate:
(a)
The position vector P
(b)
The distance vector from P to Q
(c)
The distance between P and Q
Solution to Example 3
(a) The position vector of point P
(b) The distance vector from P to Q
Solution to Example 3
(c)The distance between P and Q
Since
is a distance vector, the distance
between P and Q is the magnitude of this
distance vector.
Chapter 1 Vector Analysis
Example 4
Given points M(–1,2,1) and N(3,–3,0), find the vector RMN and
the unit vector aMN.
Solution
Multiplication of Vectors
❑ When two vectors
and
are multiplied,
the result is either a scalar or vector,
depending on how they are multiplied.
❑ Two types of multiplication:
Scalar (or dot) product
Vector (or cross) product
24
Chapter 1 Vector Analysis
The Dot Product
❑ Given two vectors A and B, the dot product, or scalar product,
❑
is defines as the prouct of the magnitude of A, the magnitude
of B, and the cosine of the smaller angle between them:
The dot product is a scalar, and it obeys the
-commutative law:
– Distributive Law
Dot Product in Cartesian
• The two vectors, and are said to be
perpendicular or orthogonal (90°) with each other
if;
Properties of dot product of unit vectors:
26
Chapter 1 Vector Analysis
Dot Product in Cartesian
■ For any vector
and
Since the angle between two unit vectors of the Cartesian
coordinate system is
, we then have:
• And thus, only three terms remain, giving finally:
Chapter 1 Vector Analysis
The Dot Product
❑ One of the most important applications of the dot product is
that of finding the component of a vector in a given direction.
• The scalar component of B in the direction of the unit vector a is B⋅a
• The vector component of B in the direction of the unit vector a is (B⋅a)a
• The geometrical term projection is also used with the dot product. Thus,
B·a is the projection of B in the a direction.
Chapter 1 Vector Analysis
The Dot Product
Example
The three vertices of a triangle are located at A(6,–1,2),
B(–2,3,–4), and C(–3,1,5). Find: (a) RAB; (b) RAC; (c) the angle
θBAC at vertex A; (d) the vector projection of RAB on RAC.
Chapter 1 Vector Analysis
The Dot Product
Example
The three vertices of a triangle are located at A(6,–1,2),
B(–2,3,–4), and C(–3,1,5). Find: (a) RAB; (b) RAC; (c) the angle
θBAC at vertex A; (d) the vector projection of RAB on RAC.
Chapter 1 Vector Analysis
The Cross Product
Given two vectors A and B, the magnitude of the cross product,
or vector product, written as A×B, is defined as the product of the
magnitude of A, the magnitude of B, and the sine of the smaller
angle between them.
Direction of
is
perpendicular (90°) to
the plane containing A
and B
Chapter 1 Vector Analysis
The Cross Product
The direction of A×B is perpendicular to the plane containing A
and B and is in the direction of advance of a right-handed screw
as A is turned into B.
Direction of
is
perpendicular (90°) to
the plane containing A
and B
The Cross Product
Chapter 1 Vector Analysis
The Cross Product
• Cross product obeys the following:
It is not commutative
It is not associative
It is distributive
Properties of Vector Product
Properties of cross product of unit vectors:
Or by using cyclic
permutation:
35
Cross product in Cartesian
• The cross product of two vectors of Cartesian
coordinate:
yields the sum of nine simpler cross products,
each involving two unit vectors.
36
Chapter 1
Cross product in Cartesian
By using the properties of cross product, it gives
and be written in more easily remembered
form:
Chapter 1 Vector Analysis
The Cross Product
Example
Given A = 2ax–3ay+az and B = –4ax–2ay+5az, find A×B.
Coordinate Systems
Chapter 1 Vector Analysis
Coordinate Systems
• Cartesian coordinates:
z
• Circular Cylindrical coordinates:
y
x
• Spherical coordinates:
40
The Cartesian Coordinate System
Chapter 1 Vector Analysis
Cartesian Coordinates
In the Cartesian coordinate system
we set up three coordinate axes
mutually at right angles to each
other and call them the x, y, and z
axes
Infinite planes: (Cartesian)
x=constant
y=constant
z=constant
• Intersection of two
planes is a line.
• Intersection of three
planes is a point.
The Cartesian Coordinate System
Chapter 1 Vector Analysis
Rectangular Coordinate System
• Differential surface units:
• Differential volume unit :
The Cylindrical Coordinate System
Chapter 1 Vector Analysis
The Cylindrical Coordinate System
The Cylindrical Coordinate System
Form by intersection of three perpendicular surfaces
or planes:
–
–
Plane of z (constant value of z)
Circular Cylinder centered on the z axis with a radius
of . Some books use the notation .
– Plane perpendicular to x-y plane and rotate about the z
axis by angle of
z
: The distance from the x-y plane to the point P in the vertical direction
: The distance from the z-axis and the point P
: The angle from x-axis toward y-axis until the projection of point P in the x-y
47
plan
Circular Cylindrical Coordinates
Note that
Circular Cylindrical Coordinates
• Unit vector of
coordinate value.
in the direction of increasing
49
Chapter 1
The Cylindrical Coordinate System
Relation between the rectangular
and the cylindrical coordinate
systems
P(x, y, z) & P(ρ ,ϕ, z)
Chapter 1
The Cylindrical Coordinate System
Dot products of unit vectors in
cylindrical and rectangular
coordinate systems
Chapter 1 Vector Analysis
The Cylindrical Coordinate System
Chapter 1 Vector Analysis
The Cylindrical Coordinate System
• Dot products of unit vectors in
cylindrical and rectangular
coordinate systems
Chapter 1 Vector Analysis
The Cylindrical Coordinate System
Dot products of unit vectors in
cylindrical and rectangular
coordinate systems
Chapter 1 Vector Analysis
The Cylindrical Coordinate System
The
components can
be obtained by
rotating the
components by
90
Chapter 1
The Cylindrical Coordinate System
Dot products of unit vectors in
cylindrical and rectangular
coordinate systems
Chapter 1 Vector Analysis
The Cylindrical Coordinate System
Dot products of unit vectors in
cylindrical and rectangular
coordinate systems
Cylindrical
coordinates
- Transformation
The Cylindrical
Coordinate
System
Chapter 1 Vector Analysis
The Cylindrical Coordinate System
Differential surface units:
Differential volume unit :
The Spherical Coordinate System
The Spherical Coordinate System
Chapter 1 Vector Analysis
The Spherical Coordinate System
Form by intersection of three perpendicular
surfaces or planes:
–
A spherical plane (constant value of r)
–
A Cone surface (constant value of θ)
–
A constant φ
plane
r: the distance from the origin to the point P
“position vector”
θ: The angle between the Z-axis and r-vector
φ : The angle between the x-axis and the point
below the point in the x-y plane
The Spherical Coordinate System
The Spherical Coordinate System
Chapter 1 Vector Analysis
The Spherical Coordinate System
Relation between the rectangular and the spherical coordinate
systems
P(x, y, z) P(r, ϴ,ϕ)
Chapter 1
The Spherical Coordinate System
Relation between the rectangular and the spherical coordinate systems
The Spherical Coordinate System
Relation between the rectangular and the spherical coordinate systems
The Spherical Coordinate System
Relation between the rectangular and the spherical coordinate systems
The Spherical Coordinate System
Relation between the rectangular and the spherical coordinate systems
Dot products of unit vectors in spherical and
rectangular coordinate systems
The Spherical Coordinate Transformation
Chapter 1 Vector Analysis
The Spherical Coordinate System
• Remember that
Chapter 1 Vector Analysis
The Spherical Coordinate System
• Differential surface units:
• Differential volume unit :
Chapter 1
The Spherical Coordinate System
Example: Given the two points, C(–3,2,1) and D(r = 5, θ = 20°,
Φ = –70°), find: (a) the spherical coordinates of C; (b) the
rectangular coordinates of D.
Summary