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. 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