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EE383 – Electromagnetics and its
Applications
Lecture 1
Introduction to Vector Analysis in Cartesian Coordinate
System
Dr. Adeem Aslam
Assistant Professor
Department of Electrical Engineering University of Engineering and Technology Lahore
Vector Analysis in Different Coordinate Systems Required to analyze vector electric and magnetic fields Review of Gauss’s Law, Divergence Theorem, Electric Potential, Boundary conditions for conductors and dielectrics, Polarization, Laplace’s and Poisson’s equations Electrostatics Review of Biot-Savart and Ampere’s Circuital Laws, Magnetic Potentials, Magnetization, Boundary conditions for magnetic materials, vector Poisson’s equation Magnetostatics Time-dependent Maxwell’s Equations, plane wave propagation and radiating elements, i.e., antennas Time-varying Electromagnetic Fields, wave propagation and antennas Grading Policy Quizzes and Assignments 15% Complex Engineering Problem (CEP) 15% Midterm Exam 30% Final Exam 40% Books Text:
- Engineering Electromagnetics, William H. Hayt, Jr. and John A. Buck, 8 th^ Ed., McGraw Hill Inc., 2012
- Field and Wave Electromagnetics, D. Cheng, 2 nd edition, Pearson Education Inc., 1998 Reference:
- Elements of Electromagnetics, Matthew N. O. Sadiku, Oxford University Press, 2006
- Electricity and Magnetism, Edward M. Purcell and David J. Morin, Cambridge University Press, 2013 Overview of the Course
- In order to analyze scalar and vector fields, rules for scalar and vector algebra must be defined first.
- Rules for scalar algebra are already well-known.
- Some of the rules for vector algebra are similar to that of scalar algebra while others differ significantly.
- The most common operation on vectors is that of addition, which follows the parallelogram law as shown in Figure 1.
- It can be observed that the vector addition is commutative, i.e.,
- Vector addition is also associative, i.e.,
Addition of Vectors
Note: A vector is represented by a bold letter and is drawn as an arrow of finite length, with the location given by its tail. Figure 1 : Parallelogram law for addition of two vectors A and. The vectors can also be added by placing the tail of B (or ) at the head of (or ). Vector Analysis
- Subtraction of a vector from is equivalent to adding the additive inverse of , i.e., , to.
- Additive inverse of a vector is obtained by reversing the direction of the vector.
- Figure 2 shows the subtraction of two vectors, which is indeed addition of a vector with the additive inverse of the other vector. Figure 2 : Parallelogram law for subtraction of two vectors. Figure 3 : Multiplication of a vector with a positive scalar changes the magnitude of the vector.
- Multiplication of a vector by a scalar changes the magnitude (length) of the vector.
- However, if the scalar is negative, the direction of the resulting vector is always reversed.
- Multiplication by a scalar also obeys the associative and distributive laws, i.e.,
- Division of a vector by a scalar is multiplication of the vector by multiplicative inverse of the scalar.
Subtraction of Vectors
Multiplication of Vectors by Scalars
Figure 3 : Multiplication of a vector with a positive scalar changes the magnitude of the vector. Vector Analysis Note : Two vectors are said to be equal if their difference is zero.
Origin Figure 6 : A right-handed rectangular/cartesian coordinate system. The curved right-hand fingers indicate the direction of turning - axis towards - axis and the direction of the thumb shows the - axis.
- Figure 6 on the right shows the simplest of coordinate systems, called the rectangular or cartesian coordinate system.
- In this system, the three coordinate axes are setup mutually perpendicular to each other.
- The axes are labeled in the right-handed convention, in which the curved fingers of the right hand indicate the rotation of - axis into - axis and the thumb shows the direction of the - axis.
- The point at which the axes meet is called the origin of the coordinate system.
- Any point in the cartesian coordinate system is represented by the - , - and - coordinates. Coordinate of a point is the distance between the origin and the intersection of the perpendicular dropped from the point to the respective axis.
- A constant coordinate plane is a surface that is perpendicular to the coordinate axis at a particular value of the coordinate. Rectangular/Cartesian Coordinate System
Figure 8 : A point is also identified as the intersection of the planes and.. So, coordinates of a point represent planes intersecting at that point. Figure 7 : Coordinates of the point are found by drawing perpendicular lines onto the respective axes. - and - coordinates of are found by dropping a perpendicular onto the plane first. Point can be thought of as the image of in the plane.
Alternative Interpretation of Coordinates
Rectangular/Cartesian Coordinate System
Figure 10 : Three unit vectors along the three mutually perpendicular axes.
- A vector in cartesian coordinate system is represented as where are the components of the vector along the respective axes and are vectors having unit magnitude along the respective coordinate axes, as shown in Figure 10 , and hence, are mutually orthogonal.
- Magnitude of a vector is a measure of its length and is given by
- Hence, the unit vector associated with , is given by dividing the vector with its magnitude, i.e., A unit vector of a vector is representative of the direction that the vector points in. Vector Representation in Cartesian Coordinate System Vector addition, subtraction and multiplication (or division) by a scalar are carried out component- wise.
- Since the coordinate unit vectors are mutually orthogonal, we observe that
- This indicates that components of a vector are projections of the vector onto respective coordinate unit vectors, i.e.,
- Dot product obeys the distributive law, hence, from equations ( 2 ) and ( 5 ), it can be represented in terms of the components of the vectors as
- Equation ( 7 ) shows that dot product of a vector with itself results in the magnitude square of the vector.
- Projection (scalar component) of a vector in the direction of , given by the unit vector , is given by where is the angle between and. Vector component of in the direction of is then given by Vector Analysis Revisited
