TEL324 — Electromagnetic Theory: Lecture Note

About this lecture note

This is the complete lecture note for TEL324 Electromagnetic Theory. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.

Preview: opening of the note

Course code: TEL 324 Course title: Electromagnetic Theory Course credit load: 2 Units Series 1: Vector Analysis and Foundations of Electromagnetic Fields • Part 1.1: Vectors and Vector Algebra o Sub Part 1.1.1: Scalars, Vectors, and Vector Representation o Sub Part 1.1.2: Vector Addition, Subtraction, and Multiplication by a Scalar o Sub Part 1.1.3: The Dot Product and Cross Product • Part 1.2: Coordinate Systems and Vector Calculus o Sub Part 1.2.1: Cartesian, Cylindrical, and Spherical Coordinate Systems o Sub Part 1.2.2: Transformation Between Curvilinear Coordinate Systems o Sub Part 1.2.3: The Gradient, Divergence, and Curl Operators • Part 1.3: Integral Calculus of Vector Fields o Sub Part 1.3.1: Line, Surface, and Volume Integrals o Sub Part 1.3.2: The Divergence Theorem o Sub Part 1.3.3: Stokes's Theorem • Part 1.4: Coulomb's Law and the Electric Field o Sub Part 1.4.1: Coulomb's Law and Electric Force o Sub Part 1.4.2: The Concept of Electric Field Intensity o Sub Part 1.4.3: Electric Field Due to Continuous Charge Distributions Introduction This is the opening Series of TEL 324: Electromagnetic Theory. Before electric and magnetic fields can be described with any rigour, learners need a solid grasp of the mathematical language in which electromagnetic theory is written, namely vector algebra and vector calculus. This

Series lays that mathematical foundation, then uses it to introduce the very first physical concept of the course: the electric field produced by stationary charges. The aim of this Series is to equip you with the vector tools, coordinate systems, and integral theorems that every later Series in this course will draw upon, and to introduce Coulomb's law and the electric field as the starting point for the study of electrostatics. We shall commence this Series by examining scalars, vectors, and the basic operations of vector algebra. Next, we will consider the Cartesian, cylindrical, and spherical coordinate systems and the vector calculus operators built upon them. Following that, we will examine the integral calculus of vector fields, including the divergence theorem and Stokes's theorem. Finally, we will conclude by examining Coulomb's law and the electric field, the first physical application of the mathematical tools developed earlier in the Series. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1: define scalar and vector quantities and represent a vector in component form, • SLO 1.2: perform vector addition, subtraction, and multiplication by a scalar, • SLO 1.3: convert vector field expressions between Cartesian, cylindrical, and spherical coordinate systems, • SLO 1.4: apply the gradient, divergence, and curl operators to vector fields, • SLO 1.5: evaluate line, surface, and volume integrals of vector fields, • SLO 1.6: apply the divergence theorem and Stokes's theorem to vector field problems, • SLO 1.7: apply Coulomb's law to determine the electric force between point charges, • SLO 1.8: calculate the electric field due to point charges and continuous charge distributions. 1.1 Vectors and Vector Algebra 1.1.1 Scalars, Vectors, and Vector Representation A scalar is a physical quantity that is completely described by a single numerical value together with an appropriate unit, such as mass, temperature, time, or electric charge. A vector, by contrast, is a quantity that possesses both magnitude and direction, such as force, velocity, or

electric field intensity. In electromagnetic theory almost every quantity of interest, from the electric field to the magnetic flux density, is a vector, so a firm grasp of vector notation and vector operations is essential before any physical law can be stated precisely. A vector is commonly written in boldface, such as A, or with an arrow above the symbol. A vector is represented in component form as A = Axax + Ayay + Azaz, where Ax, Ay, and Az are the scalar components along the x, y, and z axes and ax, ay, and az are the unit vectors, vectors of unit magnitude that indicate direction only, along each axis. The magnitude of a vector, written |A|, is obtained from the Pythagorean sum of its components, and a unit vector in the direction of A is found by dividing A by its own magnitude. Worked example: A force vector is given as F = 3ax + 4ay N. Its magnitude is |F| = √(3² + 4²) = √25 = 5 N, and the unit vector in its direction is aF = (3ax + 4ay)/5 = 0.6ax + 0.8ay. Application: Representing forces, velocities, and field intensities in component form allows an engineer to resolve a single physical quantity into independent, easily combined contributions along each coordinate direction. Fill in the blank: A scalar has magnitude only, while a vector has magnitude and ________. 1.1.2 Vector Addition, Subtraction, and Multiplication by a Scalar Two vectors are added by summing their corresponding components, so that if A = Axax + Ayay + Azaz and B = Bxax + Byay + Bzaz, then A + B = (Ax + Bx)ax + (Ay + By)ay + (Az + Bz)az. Vector subtraction follows the same rule with a sign change, A − B = A + (−B), where −B reverses the direction of B while keeping its magnitude unchanged. Both addition and subtraction obey the ordinary rules of commutativity and associativity, so the order in which several vectors are summed does not affect the final resultant. Multiplying a vector by a scalar k produces a new vector kA = kAxax + kAyay + kAzaz whose magnitude is scaled by the factor |k| and whose direction is unchanged if k is positive or reversed if k is negative. This operation is used constantly in electromagnetic theory, for instance when a field expression is multiplied by a permittivity or a charge value. Scalar multiplication distributes over vector addition, k(A + B) = kA + kB, a property that simplifies a great many field calculations. Worked example: Given A = 2ax − ay + 3az and B = ax + 4ay − az, A + B = 3ax + 3ay + 2az, and 3A = 6ax − 3ay + 9az. Application: Vector addition is how the net force, net field, or net flux at a point is found when several independent sources act simultaneously. Fill in the blank: Multiplying a vector by a negative scalar reverses its ________. 1.1.3 The Dot Product and Cross Product

The dot product (or scalar product) of two vectors A and B is defined as A·B = AxBx + AyBy + AzBz, and is equal to |A||B|cosθ, where θ is the angle between the two vectors. The result of a dot product is always a scalar, never a vector, and it provides a direct way of testing perpendicularity, since A·B = 0 whenever A and B are at right angles to one another. The dot product also gives the projection of one vector onto the direction of another, a result used repeatedly when resolving fields along a given axis. The cross product (or vector product) A × B produces a new vector whose magnitude is |A||B|sinθ and whose direction is perpendicular to both A and B, determined by the right-hand rule. In component form, A × B = (AyBz − AzBy)ax + (AzBx − AxBz)ay + (AxBy − AyBx)az. Unlike the dot product, the cross product is not commutative, since A × B = −(B × A), and it vanishes whenever A and B are parallel. Worked example: For A = ax + 2ay and B = 3ax − ay, A·B = (1)(3) + (2)(−1) = 1, and A × B = (0)ax + (0)ay + [(1)(−1) − (2)(3)]az = −7az. Application: The dot product is used to compute work done by a force and flux through a surface, while the cross product underlies the definition of torque and the magnetic force on a moving charge. Fill in the blank: The dot product of two perpendicular vectors equals ________. Figure 1.1: Vector addition by the tip-to-tail and parallelogram methods.

Pilot questions 1.1 1. Define a scalar quantity and give one example. (Linked to SLO 1.1) 2. Write the general component form of a vector A. (Linked to SLO 1.1) 3. State the formula for the magnitude of a vector A. (Linked to SLO 1.1) 4. State the rule for adding two vectors given in component form. (Linked to SLO 1.2) 5. State the formula for the dot product of two vectors A and B. (Linked to SLO 1.2) 1.2 Coordinate Systems and Vector Calculus 1.2.1 Cartesian, Cylindrical, and Spherical Coordinate Systems The Cartesian coordinate system locates a point using three mutually perpendicular axes x, y, and z, with unit vectors ax, ay, and az that are fixed in direction everywhere in space. The cylindrical coordinate system instead locates a point by a radial distance ρ from the z-axis, an azimuthal angle φ measured from the x-axis, and the height z, using unit vectors aρ, aφ, and az. Unlike the Cartesian unit vectors, aρ and aφ change direction from point to point, since they always point radially outward and tangentially around the z-axis respectively. The spherical coordinate

This is the opening of the note. Read the full note or download the PDF to continue.