Review of Mathematical Principles PDF

Biomedical Engineering Department 2nd Class College of Electronics Engineering Electromagnetic Fields Nineveh University Lecture 1 Review of Mathematical Principles Prepared By: Ass. Prof. Dr. Younis M. Abbosh Rahmah J. Abdulkareem...

Biomedical Engineering Department 2nd Class College of Electronics Engineering Electromagnetic Fields Nineveh University Lecture 1 Review of Mathematical Principles Prepared By: Ass. Prof. Dr. Younis M. Abbosh Rahmah J. Abdulkareem 1|Pa ge Introduction: Electromagnetics (EM) may be regarded as the study of the interactions between electric charges at rest and in motion. It entails the analysis, synthesis, physical interpretation, and application of electric and magnetic fields. Electromagnetics (EM) is a branch of physics or electrical engineering in which electric and magnetic phenomena are studied. EM principles find applications in various allied disciplines such as microwaves, antennas, electric machines, satellite communications, bio electromagnetic, plasmas, nuclear research, fiber optics, electromagnetic interference and compatibility, electromechanical energy conversion, radar meteorology, and remote sensing.1,2 In physical medicine, for example, EM power, in the form either of shortwaves or microwaves, is used to heat deep tissues and to stimulate certain physiological responses in order to relieve certain pathological conditions. EM fields are used in induction heaters for melting, forging, annealing, surface hardening, and soldering operations. Dielectric heating equipment uses shortwaves to join or seal thin sheets of plastic materials. EM energy offers many new and exciting possibilities in agriculture. It is used, for example, to change vegetable taste by reducing acidity. EM devices include transformers, electric relays, radio/TV, telephones, electric motors, transmission lines, waveguides, antennas, optical fibers, radars, and lasers. The design of these devices requires thorough knowledge of the laws and principles of EM. Vectors Algebra SCALARS AND VECTORS Vector analysis is a mathematical tool with which electromagnetic concepts are most conveniently expressed and best comprehended. We must learn its rules and techniques before we can confidently apply it. Since most students taking this course have little exposure to vector analysis, considerable attention is given to it in this lecture. A quantity can be either a scalar or a vector. -A scalar is a quantity that is completely specified by its magnitude. A scalar is a quantity that has only magnitude. Quantities such as time, mass, distance, temperature, entropy, electric potential, and population are scalars. -A vector has not only magnitude, but direction in space. A vector is a quantity that is described by both magnitude and direction. Vector quantities include velocity, force, momentum, acceleration displacement, and electric field intensity. - Another class of physical quantities is called tensors, of which scalars and vectors are special cases. For most of the time, we shall be concerned with scalars and vectors. *To distinguish between a scalar and a vector it is customary to represent a vector by a letter with an arrow on top of it, such as A and B, or by a letter in boldface type such as A and B. A scalar is represented simply by a letter—for example, A, B, U, and V. -EM theory is essentially a study of some particular fields. 2|Pa ge  A field is a function that specifies a particular quantity everywhere in a region. A field may indicate variation of a quantity throughout space and perhaps with time. If the quantity is scalar (or vector), the field is said to be a scalar (or vector) field. Examples of scalar fields are temperature distribution in a building, sound intensity in a theater, electric potential in a region, and refractive index of a stratified medium. And the examples of vector fields are the gravitational force on a body in space and the velocity of raindrops in the atmosphere. Unit Vector A vector A has both magnitude and direction. The magnitude of A is a scalar written as A or |A|. A unit vector aA along A is defined as a vector whose magnitude is unity (i.e., 1) and its direction is along A; that is, (1.1) Note that |aA|=1. Thus, we may write A as A = A aA (1.2) which completely specifies A in terms of its magnitude A and its direction aA. A vector A in Cartesian (or rectangular) coordinates may be represented as (Ax, Ay, Az) or Axax + Ayay + Azaz (1.3) Figure 1.1 (a) Unit vectors ax, ay, and az, (b) components of A along ax, ay, and az. The magnitude of vector A is given by (1.4) (1.5) 3|Pa ge  Vector Addition and Subtraction Two vectors A and B can be added together to give another vector C; that is, C=A+B (1.6) The vector addition is carried out component by component. Thus, if A = (Ax, Ay, Az) and B =(Bx, By, Bz). C = (Ax + Bx )ax +(Ay + By )ay + (Az + Bz )az (1.7) Vector subtraction is similarly carried out as D = A - B = A + (-B)= (Ax - Bx )ax + (Ay - By )ay + (Az – Bz )az (1.8) Solution: 4|Pa ge Hence, VECTOR MULTIPLICATION When two vectors A and B are multiplied, the result is either a scalar or a vector depending on how they are multiplied. Thus, there are two types of vector multiplication: 1. Scalar (or dot) product: A. B 2. Vector (or cross) product: A x B 5|Pa ge  A. Dot Product The dot product of two vectors A and B, written as A. B, is defined geometrically as the product of the magnitudes of A and B and the cosine of the smaller angle between them when they are drawn tail to tail. Thus, (1.9) where θAB is the smaller angle between A and B. The result of A # B is called either the scalar product because it is scalar or the dot product due to the dot sign. If A =(Ax, Ay, Az) and B =(Bx, By, Bz), then (1.10) which is obtained by multiplying A and B component by component. Two vectors A and B are said to be orthogonal (or perpendicular) with each other if A. B = 0. Note that dot product obeys the following: (i) Commutative law: A. B =B.A (1.11) (ii) Distributive law: A. (B + C) = A. B + A. C (1.12) (iii) A. A = |A|2 = A2 (1.13) Also note that ax. ay = ay. az = az. ax = 0 (1.14a) a x. ax = ay. ay = az. az = 1 (1.14b) It is easy to prove the identities in eqs. (1.11) to (1.14) by applying eq. (1.9) or (1.10). If A. B = 0, the two vectors A and B are orthogonal or perpendicular. B. Cross Product The cross product of two vectors A and B, written as A x B, is a vector quantity whose magnitude is the area of the parallelogram formed by A and B (see Figure 1.7) and is in the direction of advance of a right- handed screw as A is turned into B. (1.15) Figure 1.7 The cross product of A and B is a vector with magnitude equal to the area of the parallelogram and direction as indicated. 6|Pa ge Thus, where an is a unit vector normal to the plane containing A and B. The direction of an is taken as the direction of the right thumb when the fingers of the right hand rotate from A to B as shown in Figure 1.8(a). Alternatively, the direction of an is taken as that of the advance of a right-handed screw as A is turned into B as shown in Figure 1.8(b). The vector multiplication of eq. (1.15) is called cross product owing to the cross sign; it is also called vector product because the result is a vector. If A = (Ax, Ay, Az) and B =(Bx, By, Bz), then (1.16a) (1.16b) which is obtained by “crossing” terms in cyclic permutation, hence the name “cross product.” Figure 1.8 Direction of A x B and an using (a) the right-hand rule and (b) the right-handed-screw rule. 7|Pa ge  (1.17) (1.18) (1.19) (1.20) (1.21) (1.22) (1.23) which are obtained in cyclic permutation and illustrated in Figure 1.9. The identities in eqs. (1.17) to (1.23) are easily verified by using eq. (1.15) or (1.16). It should be noted that in obtaining an, we have used the right- hand or right-handed-screw rule because we want to be consistent with our coordinate system illustrated in Figure 1.1, which is right-handed. A right-handed coordinate system is one in which the right-hand rule is satisfied: that is, (ax x ay = az) is obeyed. In a left-handed system, we follow the left-hand or left-handed screw rule and (ax x ay =-az) is satisfied. Throughout this book, we shall stick to righthanded coordinate systems. Note: Just as multiplication of two vectors gives a scalar or vector result, multiplication of three vectors A, B, and C gives a scalar or vector result, depending on how the vectors are multiplied. Thus, we have a scalar or vector triple product. 8|Pa ge FIGURE 1.9 Cross product using cyclic permutation. (a) Moving clockwise leads to positive results. (b) Moving counterclockwise leads to negative results. Solution: Alternatively: 9|Pa ge  Coordinate Systems and Transformation In general, the physical quantities we shall be dealing with in EM are functions of space and time. In order to describe the spatial variations of the quantities, we must be able to define all points uniquely in space in a suitable manner. This requires using an appropriate coordinate system. A point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or nonorthogonal. An orthogonal system is one in which the coordinate surfaces are mutually perpendicular Nonorthogonal systems are hard to work with, and they are of little or no practical use. Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.4 A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem. A hard problem in one coordinate system may turn out to be easy in another system. In this Lecture, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical. Sometimes, it is necessary to transform points and vectors from one coordinate system to another. The techniques for doing this will be presented and illustrated with examples. 1. Cartesian Coordinates (x, y, z) point P can be represented as (x, y, z) as illustrated in Figure below. Figure 1.1 (a) Unit vectors ax, ay, and az, (b) components of A along ax, ay, and az. The ranges of the coordinate variables x, y, and z are (2.1) A vector A in Cartesian (otherwise known as rectangular) coordinates can be written as 10 | P a g e  where ax, ay, and az are unit vectors along the x-, y-, and z-directions as shown in Figure 1.1 above. The coordinate system may be either right-handed or left-handed. See Figure 1.13. It is customary to use the right- handed system. Figure 1.13 Of the rectangular coordinate systems shown in Figure above, which are not right-handed? 2. Circular Cylindrical Coordinates (ρ, φ, z) The circular cylindrical coordinate system is very convenient whenever we are dealing with problems having cylindrical symmetry, such as dealing with a coaxial transmission line. A point P in cylindrical coordinates is represented (ρ, φ, z)and is as shown in Figure 2.1. Observe Figure 2.1 closely and note how we define each space variable: ρ is the FIGURE 2.1 Point P and unit vectors in the cylindrical coordinate system. 11 | P a g e 12 | P a g e 13 | P a g e  3. Spherical Coordinates (r, θ, φ) Although cylindrical coordinates are covered in calculus texts, the spherical coordinates are rarely covered. The spherical coordinate system is most appropriate when one is dealing with problems having a degree of spherical symmetry. A point P can be represented 14 | P a g e 15 | P a g e 16 | P a g e For the sake of completeness, it may be instructive to obtain the point or vector transformation relationships between cylindrical and spherical coordinates. We shall use Figures 2.5 and 2.6 (where ϕ is held constant, since it is common to both systems). Note that in a point or vector transformation, the point or vector has not changed; it is only expressed differently. Thus, for example, the magnitude of a vector will remain the same after the transformation, and this may serve as a way of checking the result of the transformation. The distance between two points is usually necessary in EM theory. The distance d between two points with position vectors r1 and r2 is generally given by 17 | P a g e Solution: 18 | P a g e 19 | P a g e 20 | P a g e ‫ماما‬ 21 | P a g e 22 | P a g e  Vector Calculus ❖ DEL OPERATOR 23 | P a g e ❖ GRADIENT OF A SCALAR The gradient of a scalar field at any point is the maximum rate of change of the field at that point The gradient of a scalar field V is a vector that represents both the magnitude and the direction of the maximum space rate of increase of V. A mathematical expression for the gradient can be obtained by evaluating the difference in the field dV between points P1 and P2 of Figure 3.13, where V1, V2, and V3 are contours on which V is constant. From calculus, 24 | P a g e or 25 | P a g e The following computation formulas on gradient, which are easily proved, should be noted: where U and V are scalars and n is an integer. Also take note of the following fundamental properties of the gradient of a scalar field V: 1. The magnitude of ▽V equals the maximum rate of change in V per unit distance. 2. ▽V points in the direction of the maximum rate of change in V. 3. ▽V at any point is perpendicular to the constant V surface that passes through that point (see points P and Q in Figure 3.13(in page25)). 4. The projection (or component) of ▽V in the direction of a unit vector a is =V. a and is called the directional derivative of V along a. This is the rate of change of V in the direction of a. For example, dV/dl in eq. (3.26) is the directional derivative of V along P1P2 in Figure 3.13. Thus the gradient of a scalar function V provides us with both the direction in which V changes most rapidly and the magnitude of the maximum directional derivative of V. 5. If A = V, V is said to be the scalar potential of A. 26 | P a g e ❖ Divergence of A Vector and Divergence Theorem The divergence of A at a given point P is the outward flux per unit volume as the volume shrinks about P. Hence, 27 | P a g e The divergence of A at point P (xo, yo, zo) in a Cartesian system is given by we obtain the divergence of A in cylindrical coordinates as we obtain the divergence of A in spherical coordinates as 28 | P a g e 29 | P a g e  ❖ CURL OF A VECTOR The curl of A is an axial (or rotational) vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the circulation maximum. Because of its rotational nature, some authors use rot A instead of curl A. 30 | P a g e 31 | P a g e 32 | P a g e 33 | P a g e 34 | P a g e