Vector Analysis and Coordinate Systems 2023/2024 PDF

Vector Analysis and Coordinate Systems Course: Electromagnetic Fields Academic Year: 2023/2024 Lecturer: Dr. Mohamed A. El-Shimy Textbook: William Hayt, ‘Engineering Electromagnetics’, Ed. 8. 1. Vector Definition 1.1. Scalar and Vector  Scalar – A quantity that is specified by a real number. (E...

Vector Analysis and Coordinate Systems Course: Electromagnetic Fields Academic Year: 2023/2024 Lecturer: Dr. Mohamed A. El-Shimy Textbook: William Hayt, ‘Engineering Electromagnetics’, Ed. 8. 1. Vector Definition 1.1. Scalar and Vector  Scalar – A quantity that is specified by a real number. (Ex: volume, charge)  Vector – A quantity that has direction and magnitude. The direction is represented by an arrow and its magnitude is the length of the arrow. (Ex: electric and magnetic field intensities) 1.2. Unit Vector  It is a vector of a magnitude 1. The unit vector in the direction of a vector 𝐀 : 𝑨 𝒂𝑨 = |𝑨|  In a 3-D coordinate system of unit vectors (𝒂𝟏 , 𝒂𝟐 , 𝒂𝟑 ), a vector 𝐀 can be represented in a component form: 𝐀 = 𝐴 𝐚𝟏 + 𝐴 𝐚𝟐 + 𝐴 𝐚𝟑 and its magnitude |𝐀| = 𝐴 +𝐴 +𝐴 2. Vector Algebra 2.1. Addition and Subtraction  Vectors can be added and subtracted. 𝐂 = 𝐀 ± 𝐁 = (𝐴 ± 𝐵 )𝐚𝟏 + (𝐴 ± 𝐵 )𝐚𝟐 + (𝐴 ± 𝐵 )𝐚𝟑 2.2.Dot or Scalar Product  Dot product of two vectors 𝐀 and 𝐁: 𝐀 ⋅ 𝐁 = |𝐀||𝐁| cos 𝜃 where 0 ≤ 𝜃 ≤ 𝜋 is the smaller angle between 𝐀 and 𝐁.  In component form: 𝐀⋅𝐁=𝐴 𝐵 +𝐴 𝐵 +𝐴 𝐵  Properties  It is a scalar value. Then 𝐀 ⋅ 𝐁 = 𝐁 ⋅ 𝐀.  The projection of 𝐀 in a certain direction of unit vector 𝐚𝐁 : 𝐀 ⋅ 𝐚𝐁 = |𝐀| cos 𝜃  The dot product of vector by itself: 𝐀 ⋅ 𝐀 = |𝐀|𝟐  Two orthogonal vectors 𝐀 ⊥ 𝐁 : 𝐀 ⋅ 𝐁 = 0. 2.3.Cross or Vector Product  Cross products of two vectors 𝐀 and 𝐁: 𝐀 × 𝐁 = |𝐀||𝐁| sin 𝜃 𝒂𝒏 where 0 ≤ 𝜃 ≤ 𝜋 is the smaller angle between 𝐀 and 𝐁, and 𝒂𝒏 is a unit vector ⊥ to the plane of 𝐀 and 𝐁 (right-handed screw).  In component form: 𝒂𝟏 𝒂𝟐 𝒂𝟑 𝐀×𝐁 = 𝐴 𝐴 𝐴 𝐵 𝐵 𝐵 Properties  Area of parallelogram = |𝐀 × 𝐁|  Cross-product of vector by itself: 𝐀 × 𝐀 = 𝟎  Two parallel vectors 𝐀||𝐁: 𝐀 × 𝐁 = 𝟎 Example 1 Given the vector 𝐄 = 2 𝐚𝐱 + 4 𝐚𝐲 at the origin, find: a) The unit vector along 𝐄. b) The angle between the vector 𝐄 and the positive x-axis. c) A vector 𝐇 that is perpendicular to the vector 𝐄. Ans: a) 𝐚𝐄 = 𝐄/|𝐄|, where |𝐄| = 2√5 b) 𝜙 = cos (𝐚𝐄 ⋅ 𝐚𝐱 ) = 63.4 3. Orthogonal Coordinate Systems 3.1.Cartesian or Rectangular Coordinates (𝒙, 𝒚, 𝒛)  It is the simplest coordinate system defined by three orthogonal coordinate axes (𝑥, 𝑦, 𝑧) where −∞ < 𝑥 < ∞, −∞ < 𝑦 < ∞, and −∞ < 𝑧 < ∞. Coordinates of a Point  A point 𝑃 is located by the intersection of three mutually perpendicular surfaces on which one of the coordinate point values is constant: (1) 𝑥 = constant ⇒ plane at 𝑥 // 𝑦𝑧-plane (2) 𝑦 = constant ⇒ plane at 𝑦 // 𝑥𝑧-plane (3) 𝑧 = constant ⇒ plane at 𝑧 // 𝑥𝑦-plane  Unit Coordinate Vectors  Three-unit vectors (𝐚𝐱 , 𝐚𝐲 , 𝐚𝐳 ) at point 𝑃 are defined where each is pointed along one of the coordinate axes in the direction of increasing coordinate values.  The unit vectors have fixed directions independent of the location of 𝑃.  Typically, a right-handed coordinate system is defined where 𝐚𝐱 × 𝐚𝐲 = 𝐚𝐳.  Vector in the rectangular component form: 𝐀 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 (a) Coordinates of 𝑃(𝑥, 𝑦, 𝑧) (b) Perpendicular surfaces (c) Unit coordinate vectors 3.2. Circular Cylindrical Coordinates (𝝆, 𝝓, 𝒛)  Many problems possess a type of cylindrical symmetry which is easier to be handled with cylindrical instead of rectangular coordinates.  Cylindrical coordinate system (𝜌, 𝜙, 𝑧) is a 3-D version of the polar coordinates (𝜌, 𝜙) by specifying a third dimension 𝑧 of a point’s location.  Coordinates parameters ranges are 𝜌 ≥ 0, 0 ≤ 𝜙 < 2𝜋, and −∞ < 𝑧 < ∞. Coordinates of a Point  Similarly, in cylindrical coordinates, a point 𝑃 is located by the intersection of three mutually perpendicular surfaces on which one of the coordinate values is constant: (1) 𝜌 = constant ⇒ circular cylinder, (2) 𝜙 = constant ⇒ half-plane (edge along 𝑧 axis) ⊥ 𝑥𝑦-plane, (3) 𝑧 = constant ⇒ infinite plane // 𝑥𝑦-plane. Unit Coordinate Vectors  Three mutually perpendicular unit vectors (𝐚𝛒 , 𝐚𝛟 , 𝐚𝐳 ) at point 𝑃 are defined to be directed toward increasing coordinate values and each is normal to its corresponding perpendicular surface.  The unit vectors depend on the location of 𝑃 (except for 𝐚𝐳 ).  The system is defined as a right-handed one: 𝐚𝛒 × 𝐚𝛟 = 𝐚𝐳  Vector in the cylindrical component form: 𝐀 = 𝐴 𝐚𝛒 + 𝐴 𝐚𝛟 + 𝐴 𝐚𝐳  (a) Coordinates of 𝑃(𝜌, 𝜙, 𝑧) (b) Perpendicular surfaces (c) Unit coordinate vectors 3.3. Spherical Coordinates (𝒓, 𝜽, 𝝓)  A problem has spherical symmetry would be easy solved in spherical coordinates.  Spherical coordinate (𝑟, 𝜃, 𝜙) is a 3-D version of the polar coordinates (𝜌, 𝜙).  Coordinates parameters ranges are 𝑟 ≥ 0, 0 ≤ 𝜃 < 𝜋, and 0 ≤ 𝜙 < 2𝜋. Coordinates of a Point  In spherical coordinates, a point 𝑃 is located by the intersection of three mutually perpendicular surfaces on which one of the coordinate values is constant: (1) 𝑟 = constant ⇒ sphere centered at the origin, (2) 𝜃 = constant ⇒ circular cone with vertex at the origin and axis in the 𝑧-axis, (3) 𝜙 = constant ⇒ half-plane (edge along 𝑧 axis) ⊥ 𝑥𝑦-plane. Unit Coordinate Vectors  Three mutually perpendicular unit vectors (𝐚𝐫 , 𝐚𝛉 , 𝐚𝛟 ) at point 𝑃 are defined to be directed toward increasing coordinate values and each is normal to one of the three perpendicular surfaces. They depend on the location of 𝑃.  The coordinate system is defined as 𝐚𝐫 × 𝐚𝛉 = 𝐚𝛟  Vector in the spherical component form: 𝐀 = 𝐴 𝐚𝒓 + 𝐴 𝐚𝛉 + 𝐴 𝐚𝛟 (a) Coordinates of 𝑃(𝑟, 𝜃, 𝜙) (b) Perpendicular surfaces (c) Unit coordinate vectors  Example 2 Given the vector 𝐄 = 2 𝐚𝐱 + 4 𝐚𝐲 at point (1,2,0). Express the vector 𝐄 in a) cylindrical coordinates b) spherical coordinates. Ans: a) 𝐄 = 2√5 𝐚𝛒 , at 𝜌 = √5, 𝜙 = 63.4 , and 𝑧 = 0 b) 𝐄 = 2√5 𝐚𝐫 , at 𝑟 = √5, 𝜃 = 90.0 , and 𝜙 = 63.4 4. Coordinate Transformation 4.1. Cylindrical-Rectangular Transformation Variable Transformation  The variables of the rectangular and cylindrical coordinate systems are related to each other by: Cylindrical → Rectangular Rectangular → Cylindrical 𝑥 = 𝜌 cos 𝜙 𝜌= 𝑥 +𝑦 𝑦 𝑦 = 𝜌 sin 𝜙 𝜙 = tan 𝑥 𝑧=𝑧 𝑧=𝑧 Vector Transformation  Given a rectangular vector 𝐀 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 and a cylindrical vector 𝐀 = 𝐴 𝐚𝛒 + 𝐴 𝐚𝛟 + 𝐴 𝐚𝐳  The components of a cylindrical vector: o 𝐴 = 𝐀 ⋅ 𝐚𝛒 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 ⋅ 𝐚𝛒 o 𝐴 = 𝐀 ⋅ 𝐚𝛟 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 ⋅ 𝐚𝛟 o 𝐴 = 𝐀 ⋅ 𝐚𝐳 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 ⋅ 𝐚𝐳  The results of the dot products are given in the following table. Rectangular 𝐚𝐱 𝐚𝐲 𝐚𝐳 𝐚𝛒 cos 𝜙 sin 𝜙 0 Cylindrical 𝐚𝛟 − sin 𝜙 cos 𝜙 0 𝐚𝐳 0 0 1 4.2. Spherical-Rectangular Transformation  The variables of the rectangular and spherical coordinate systems are related to each other by: Spherical → Rectangular Rectangular → Spherical 𝑥 = 𝑟 sin 𝜃 cos 𝜙 𝑟= 𝑥 +𝑦 +𝑧 𝑧 𝑦 = 𝑟 sin 𝜃 sin 𝜙 𝜃 = cos 𝑥 +𝑦 +𝑧 𝑦 𝑧 = 𝑟 cos 𝜃 𝜙 = tan 𝑥 Vector Transformation  Given a rectangular vector 𝐀 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 and a spherical vector 𝐀 = 𝐴 𝐚𝐫 + 𝐴 𝐚𝛉 + 𝐴 𝐚𝛟  The components of a cylindrical vector: o 𝐴 = 𝐀 ⋅ 𝐚𝐫 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 ⋅ 𝐚𝐫 o 𝐴 = 𝐀 ⋅ 𝐚𝛉 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 ⋅ 𝐚𝛉 o 𝐴 = 𝐀 ⋅ 𝐚𝛟 = 𝐴 𝐚𝐱 + 𝐴 𝐚𝐲 + 𝐴 𝐚𝐳 ⋅ 𝐚𝛟  The results of the dot products are given in the following table. Rectangular 𝐚𝐱 𝐚𝐲 𝐚𝐳 𝐚𝐫 sin 𝜃 cos 𝜙 sin 𝜃 sin 𝜙 cos 𝜃 Spherical 𝐚𝛉 cos 𝜃 cos 𝜙 cos 𝜃 sin 𝜙 − sin 𝜃 𝐚𝛟 − sin 𝜙 cos 𝜙 0 Example 3 Using coordinate transformation, express the vector 𝐄 = 2 𝐚𝐱 + 4 𝐚𝐲 at point (1,2,0) into cylindrical and spherical coordinates. Ans: See Example 2 5. Differential Elements  There are some problems that have to be solved through integration along a curve, over a surface, or throughout a volume.  Increase each coordinate values of a point 𝑃 by a differential amount, a differential element (line, surface, or volume) is formed. Rectangular  Point 𝑃′: 𝑥 + 𝑑𝑥, 𝑦 + 𝑑𝑦, 𝑧 + 𝑑𝑧  Differential Line: 𝒅𝒍 = 𝑑𝑥 𝐚𝐱 + 𝑑𝑦 𝐚𝐲 + 𝑑 𝐚𝐳  Differential Area 𝐝𝐒 o Constant 𝑥: 𝐝𝐒𝐱 = 𝑑𝑦 𝑑𝑧 𝐚𝐱 o Constant 𝑦: 𝐝𝐒𝐲 = 𝑑𝑥 𝑑𝑧 𝐚𝐲 o Constant 𝑧: 𝐝𝐒𝐳 = 𝑑𝑥 𝑑𝑦 𝐚𝐳  Differential Volume: 𝑑𝑣 = 𝑑𝑥 𝑑𝑦 𝑑𝑧 Cylindrical  Point 𝑃′: 𝜌 + 𝑑𝜌, 𝜙 + 𝑑𝜙, 𝑧 + 𝑑𝑧  Differential Line: o In 𝜌 direction: 𝐝𝐥𝛒 = 𝑑𝜌 𝐚𝛒 o In 𝜙 direction: 𝐝𝐥𝛟 = 𝜌𝑑𝜙 𝐚𝛟 o In 𝑧 direction: 𝐝𝐥𝐳 = 𝑑𝑧 𝒂𝒛 o 𝒅𝒍 = 𝑑𝜌 𝐚𝛒 + 𝜌𝑑𝜙 𝐚𝛟 + 𝑑𝑧 𝒂𝒛  Differential Area 𝐝𝐒 o Constant 𝜌: 𝐝𝐒𝛒 = 𝜌 𝑑𝜙 𝑑𝑧 𝐚𝛒 o Constant 𝜙: 𝐝𝐒𝛟 = 𝑑𝜌 𝑑𝑧 𝐚𝛟 o Constant 𝑧: 𝐝𝐒𝐳 = 𝜌 𝑑𝜙 𝑑𝜌 𝐚𝐳 Differential Volume: 𝑑𝑣 = 𝜌 𝑑𝜌 𝑑𝜙 𝑑𝑧 Spherical  Point 𝑃′: 𝑟 + 𝑑𝑟, 𝜃 + 𝑑𝜃, 𝜙 + 𝑑𝜙  Differential Line: o In 𝑟 direction: 𝐝𝐥𝐫 = 𝑑𝑟 𝐚𝐫 o In 𝜃 direction: 𝐝𝐥𝛉 = 𝑟𝑑𝜃 𝐚𝛉 o In 𝜙 direction: 𝐝𝐥𝛟 = 𝑟 sin 𝜃 𝑑𝜙 𝒂𝝓 o 𝒅𝒍 = 𝑑𝑟 𝐚𝐫 + 𝑟𝑑𝜃 𝐚𝛉 + 𝑟 sin 𝜃 𝑑𝜙 𝒂𝝓  Differential Area 𝐝𝐒 o Constant 𝑟: 𝐝𝐒𝐫 = 𝑟 sin 𝜃 𝑑𝜃 𝑑𝜙 𝐚𝐫 o Constant 𝜃: 𝐝𝐒𝛉 = 𝑟 sin 𝜃 𝑑𝑟 𝑑𝜙 𝐚𝛉 o Constant 𝜙: 𝐝𝐒𝛟 = 𝑟 𝑑𝑟 𝑑𝜃 𝐚𝛟 Differential Volume: 𝑑𝑣 = 𝑟 sin 𝜃 𝑑𝑟 𝑑𝜃 𝑑𝜙 Example 4 Starting from a differential element of a sphere of radius 𝑎, obtain an expression for a) The surface area of the sphere b) The volume of the sphere Ans: a) 4𝜋𝑎 b) 𝜋𝑎  Rectangular Cylindrical Spherical Point 𝑷 to 𝑷′ 𝑃(𝑥, 𝑦, 𝑧) → 𝑃(𝑥 + 𝑑𝑥, 𝑦 + 𝑑𝑦, 𝑧 + 𝑑𝑧) 𝑃(𝜌, 𝜙, 𝑧) → 𝑃(𝜌 + 𝑑𝜌, 𝜙 + 𝑑𝜙, 𝑧 + 𝑑𝑧) 𝑃(𝑟, 𝜃, 𝜙) → 𝑃(𝑟 + 𝑑𝑟, 𝜃 + 𝑑𝜃, 𝜙 + 𝑑𝜙) Differential Line o In 𝑥 direction: 𝐝𝐥 𝐱 = 𝑑𝑥 𝐚𝐱 o In 𝜌 direction: 𝐝𝐥𝛒 = 𝑑𝜌 𝐚𝛒 o In 𝑟 direction: 𝐝𝐥𝐫 = 𝑑𝑟 𝐚𝐫 o In 𝑦 direction: 𝐝𝐥 𝐲 = 𝑑𝑦 𝐚𝐲 o In 𝜙 direction: 𝐝𝐥𝛟 = 𝜌 𝑑𝜙 𝐚𝛟 o In 𝜃 direction: 𝐝𝐥𝛉 = 𝑟 𝑑𝜃 𝐚𝛉 o In 𝑧 direction: 𝐝𝐥𝐳 = 𝑑𝑧 𝒂𝒛 o In 𝑧 direction: 𝐝𝐥𝐳 = 𝑑𝑧 𝒂𝒛 o In 𝜙 direction: 𝐝𝐥𝛟 = 𝑟 sin 𝜃 𝑑𝜙 𝒂𝝓 o 𝐝𝐥 = 𝑑𝑥 𝐚𝐱 + 𝑑𝑦 𝐚𝐲 + 𝑑 𝐚𝐳 o 𝐝𝐥 = 𝑑𝜌 𝐚𝛒 + 𝜌 𝑑𝜙 𝐚𝛟 + 𝑑𝑧 𝒂𝒛 o 𝐝𝐥 = 𝑑𝑟 𝐚𝐫 + 𝑟 𝑑𝜃 𝐚𝛉 + 𝑟 sin 𝜃 𝑑𝜙 𝒂𝝓 Differential Area o Constant 𝑥: 𝐝𝐒𝐱 = 𝑑𝑦 𝑑𝑧 𝐚𝐱 o Constant 𝜌: 𝐝𝐒𝛒 = 𝜌 𝑑𝜙 𝑑𝑧 𝐚𝛒 o Constant 𝑟: 𝐝𝐒𝐫 = 𝑟 sin 𝜃 𝑑𝜃 𝑑𝜙 𝐚𝐫 o Constant 𝑦: 𝐝𝐒𝐲 = 𝑑𝑥 𝑑𝑧 𝐚𝐲 o Constant 𝜙: 𝐝𝐒𝛟 = 𝑑𝜌 𝑑𝑧 𝐚𝛟 o Constant 𝜃: 𝐝𝐒𝛉 = 𝑟 sin 𝜃 𝑑𝑟 𝑑𝜙 𝐚𝛉 o Constant 𝑧: 𝐝𝐒𝐳 = 𝑑𝑥 𝑑𝑦 𝐚𝐳 o Constant 𝑧: 𝐝𝐒𝐳 = 𝜌 𝑑𝜙 𝑑𝜌 𝐚𝐳 o Constant 𝜙: 𝐝𝐒𝛟 = 𝑟 𝑑𝑟 𝑑𝜃 𝐚𝛟 Differential Volume 𝑑𝑣 = 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝑣 = 𝜌 𝑑𝜌 𝑑𝜙 𝑑𝑧 𝑑𝑣 = 𝑟 sin 𝜃 𝑑𝑟 𝑑𝜃 𝑑𝜙