A_Course_Manual_on_Engineering_Electroma.

1 A Course Manual on Engineering Electromagnetics
A_Course_Manual_on_Engineering_Electroma

Chapter 1
Introduction
1.1. Scalars and Vectors
Any quantity can be either scalar or vector. The term scalar refers to a quantity whose value may be
represented by a single real number. Scalar quantity has magnitude only. Physical quantities such as mass,
density, distance, pressure, temperature, volume, time etc are scalars. Voltage is also a scalar quantity.
A vector quantity has both magnitude and direction. Physical quantities such as force, velocity, acceleration
etc are vector quantities. Vector quantities are represented by letters with overhead arrow. The unit vector
of a vector quantity A is represented by
→
A
a = which has a unit magnitude along the direction of A.
A
In rectangular coordinate system: ax , a y and a z are the unit vectors along x, y and z-axes respectively. Let a
vector A have components Ax, Ay and Az along x, y and z-axes respectively. Then,
A= A2x + A2y + A2z


1.2. Vector Algebra
The addition of vector follows parallelogram law.
A+B=B+A (Commutative)
A + (B + C) = (A + B) + C (Associative)




Figure 1.1: Two vectors may be added graphically either by drawing parallelogram or triangle.


The vector should be coplanar (vectors lying in a common plane) to obey these laws. The subtraction of
vectors A and B is equivalent to reversing the vector B and adding to A.
Any vector A multiplied by scalar quantity k gives the vector kA of magnitude k A along the direction of A.
But the multiplication of two vectors results

Scalar Product
Scalar product (dot product) of two vectors A and B is represented as
A.B = A B cosθ where  is the angle between vector A and B.
The dot product follows:


Prepared by: Ashok Chhetry, Himalaya College of Engineering, Lalitpur, Nepal

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A.B = B.A (Commutative)
A.(B + C) = A.B + A.C (Distributive)
The dot product of two vectors is scalar quantity.
When  = 90o , A.B = 0 (i.e. two vectors are said to be perpendicular if their dot product is zero.)


Vector Product
Vector product of two vectors A and B are represented by A B, the magnitude of cross product is
A  B = A B sin â N . The direction of A  B is perpendicular to the plane containing A and B (given by
right hand rule). The vector product follows:
A B = −B  A (Non-commutative)
A (B + C) = A  B + A  C (Distributive)
A (B)  ( A  B)  C (Non-associative)
When  = 0o A  B = 0 (i.e. two vectors are said to be parallel if their cross product is zero.)


1.3. Coordinate System
Although law of electromagnetism are invariant with coordinate system. Solutions of practical problems
require that the relations derived from these laws are expressed in coordinate system. Three coordinate
systems are:
1. Rectangular Cartesian coordinate system
In rectangular coordinate system we set up three coordinate axes mutually at right angles to each other
and call them the x, y and z axes. Any point in the space is considered as common intersection of three
planes x = constant, y = constant and z = constant.

2. Circular Cylindrical coordinate System
Any point in the space is considered as a point of intersection of three mutually orthogonal surfaces
a) circular cylindrical surface of constant radius, 
b) angle  by shifting xz-plane and
c) constant z, parallel to xy-plane
The variables of rectangular and cylindrical coordinate system are easily related to each other as
x =  cos , y =  sin and z = z .
These are the transformation equations from cylindrical to rectangular Cartesian coordinate system.




Prepared by: Ashok Chhetry, Himalaya College of Engineering, Lalitpur, Nepal

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Figure 1.2: a) Right handed rectangular coordinate system, b) The location of point P(1,2,3) and Q(2, -2, 1) and c)
The differential volume element




Figure 1.3: a) Three mutually perpendicular surfaces of circular cylindrical coordinate system b) Three unit vectors
and c) Differential volume element.
Thus, inverse transformation equations are related as:

 = x2 + y2 (  0) and

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= −1
y (0    2 )
tan  
x
A vector A in rectangular coordinate system is expressed as A = Ax ax + Ay ay + Az az and in circular
cylindrical coordinate system as A = Ax a + Ay a + Az az . A component of a vector in desired direction
may be obtained by taking the dot product of that vector with the unit vector in the desired direction. Hence,
A = Ax ax .a + Ay ay .a + Az az .a

A = Ax ax .a + Ay ay .a + Az az .a and Az = Ax ax .az + Ay ay .az + Az az .az
In order to complete the transformations of the component it is necessary to know relationships between dot
products of unit vectors in rectangular coordinate system with cylindrical coordinate system. The dot
products of unit vectors are tabulated as:

a a az

ax . cos − sin 0
ay . sin cos 0
az . 0 0 1


3. Spherical coordinate System
Any point in the space is considered as a point of intersection of three mutually perpendicular surfaces
a) constant radius from origin, r
b) angle  with z-axis and
c) angle  with x-axis (identical to that of cylindrical coordinate system)
The transformation equations from spherical to rectangular Cartesian coordinate system are described by
equations:
x = r sin cos , y = r sin sin and z = r cos
The transformation in reverse direction is achieved through,

r= x2 + y2 + z 2 (r  0)

 = −1 
y (0    2 ) and
tan  
x
 z 
 = cos −1  (0     )
 2 
 x + y +z 
2 2




Prepared by: Ashok Chhetry, Himalaya College of Engineering, Lalitpur, Nepal