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Chapter One 1.1 Vector analysis Vector Representation This introductory chapter is a review of mathematical concepts required for the electromagnetism. It is assumed that the reader is already familiar with elementary vector analysis. Physical quantities that we deal with in electromagnetism can be scalars or vectors. A scalar is an entity which only has a magnitude. Examples of scalars are mass, time, distance, electric charge, electric potential, energy, temperature etc. A vector is characterized by both magnitude and direction. Examples of vectors in physics are displacement, velocity, acceleration, force, electric ﬁeld, magnetic ﬁeld etc. A vector is represented by a straight line of specific length having an arrow head which shows its direction. This straight line is called the representative line of that vector. Vector variables are sometimes notated with arrow above them (particularly in handwritten physics formulae), and sometimes notated by using a bold-face font (particularly in print). A ﬁeld is a quantity which can be speciﬁed everywhere in space as a function of position. The quantity that is speciﬁed may be a scalar or a vector. For instance, we can specify the temperature at every point in a room. The room may, therefore, be said to be a region of “temperature ﬁeld” which is a scalar ﬁeld because the temperature T(x, y, z) is a scalar function of the position. An example of a scalar ﬁeld in electromagnetism is the electric potential. In a similar manner, a vector quantity which can be speciﬁed at every point in a region of space is a vector ﬁeld. For instance, every point on the earth may be considered to be in the gravitational force ﬁeld of the earth. We may specify the ﬁeld by the magnitude and the direction of acceleration due to gravity (i.e. force per unit mass) g(x, y, z) at every point in space. As another example consider ﬂow of water in a pipe. At each point in the pipe, the water molecule has a velocity ⃗ (x, y, z). The water in the pipe may be said to be in a velocity ﬁeld. There are several examples of vector ﬁeld in electromagnetism, e.g., the electric ﬁeld ⃗⃗ , the magnetic ﬂux density ⃗⃗ etc. 1 Chapter One Vector analysis Orthogonal coordinate systems Rectangular (Cartesian) Coordinates P (x, y, z) In Cartesian (rectangular) coordinates a point P is specified by x, y, and z, where these values are all measured from the origin (see figure at right). Cylindrical Coordinates P (r, Φ, z) In cylindrical coordinates a point P is specified by r, Φ, z, where Φ is measured from the x axis (see figure at right). Spherical Coordinates P (r, θ, Φ) In spherical coordinates a point P is specified by r, θ, Φ, where r is measured from the origin, θ is measured from the z axis, and Φ is measured from the x axis (see figure at right). 1.2 Cartesian Coordinates A vector at the point P is specified in terms of three mutually perpendicular components with unit vectors 𝑥̂, 𝑦̂, 𝑎𝑛𝑑 𝑧̂ , (also called 𝑖̂, 𝑗̂ and 𝑘̂, or aˆ x, aˆ y , and aˆ z ). Vector representation A Axaˆ x Ayaˆ y Azaˆ z or A xˆ A x yˆ A y zˆ A z Magnitude of A A A A A 2x A 2y A 2z Position vector A xˆ x1 yˆ y1 zˆ z1 2 Chapter One Vector analysis Base vector properties xˆ xˆ yˆ yˆ zˆ zˆ 1 xˆ yˆ yˆ zˆ zˆ xˆ 0 xˆ yˆ zˆ yˆ zˆ xˆ zˆ xˆ yˆ Dot product: A B AxBx Ay By Az Bz Cross product: xˆ A B Ax yˆ zˆ Ay Az Bx By Bz 1.3 Cylindrical Coordinates Cylindrical representation uses: r, Φ, z A Araˆ r Aaˆ Azaˆ z A B ArBr AB AzBz Dot Product (Scalar) aˆ Unit Vectors 1.4 r aˆ aˆ z Spherical coordinates Spherical representation uses: r, θ, Ф A Ar aˆ r A aˆ A aˆ A B ArBr AB AB aˆ r aˆ aˆ Dot Product (Scalar) Unit Vectors 3 Chapter One Vector analysis Vector Representation: Unit Vectors Rectangular Coordinate System Unit Vector Representation for Rectangular Coordinate System The Unit Vectors imply: â x Points in the direction of increasing x â y Points in the direction of increasing y â z Points in the direction of increasing z Cylindrical Coordinate System The Unit vectors imply: â r Points in the direction of increasing r â Points in the direction of increasing Ф â z Points in the direction of increasing z Spherical Coordinate System The Unit Vectors imply: â r Points in the direction of increasing r â Points in the direction of increasing θ â Points in the direction of increasing Ф 4 Chapter One 1.5 Vector analysis Differential length, area, and volume 1- Cartesian Coordinates When you move a small amount in x-direction, the distance is dx. In a similar fashion, you generate dy and dz. Differential quantities: Length (displacement): d l xˆ dx yˆ dy zˆ dz Area (normal area): d sx xˆ dydz d sy yˆ dxdz d sz zˆ dxdy Volume: dv dxdydz 2- Cylindrical Coordinates ( r, Φ, z): 0r r radial distance in x-y plane Φ azimuth angle measured from the positive x-axis Z z Vector representation ˆ A zˆ A z A aˆ A rˆA r Magnitude of A A A A A r2 A 2 A 2z Position vector A rˆr1 zˆ z1 5 0 2 Chapter One Vector analysis Base vector properties ˆ zˆ , rˆ ˆ zˆ rˆ , ˆ zˆ rˆ Dot product: A B Ar Br A B A z B z Cross product: rˆ A B Ar ˆ A zˆ Az Br B Bz Length (displacement): ˆ rd zˆ dz d l rˆdr Area (normal area): d sr rˆrd dz ˆ drdz d s d sz zˆ rdrd Volume: dv rdrd dz 3- Spherical Coordinates (R, θ, Φ) Vector representation ˆ A R ˆ A ˆ A AR Magnitude of A A A A A 2R A 2 A 2 6 Chapter One Vector analysis Position vector A ˆ R1 R Base vector properties ˆ ˆ ˆ, R ˆ ˆ, ˆ R ˆ ˆ ˆ R Dot product: A B A R B R A B A B Cross product: ˆ R A B AR ˆ A ˆ A BR B B Differential quantities: Length (displacement): ˆ dl R ˆ dl ˆ dl dl R ˆ dR ˆ Rd ˆ R sin d R Area (normal area): ˆ dl dl R ˆ R 2 sin dd d sR R ˆ dl R dl ˆ R sin dRd d s ˆ dl R dl ˆ RdRd d s Volume: dv R 2 sin dRdd dl R dR dl Rd dl R sin d 7 Chapter One Vector analysis Cartesian to Cylindrical Transformation A r A x cos A y sin A A x sin A y cos Az Az r x2 y 2 tan 1 (y / x) [ zz rˆ xˆ cos yˆ sin ˆ xˆ sin yˆ cos zˆ zˆ The fundamental parameters of the rectangular, cylindrical, and spherical coordinate systems are summarized in the following table: 8