Electromagnetic fields: Review of vector algebra

EC8451- ELECTROMAGNETIC FIELDS
Dr.K.G.SHANTHI
Professor/ECE
shanthiece@rmkcet.ac.in
Review of vector algebra-Problems
•Unit Vector, Position Vector, Distance vector
•Vector Addition
•Multiplication of a Vector by a Scalar
•Scalar or dot product: A • B
•Vector or cross product: A x B
Orthogonal Coordinate Systems-Introduction

✘ Electromagnetics is a branch of physics or electrical engineering that
deals electric and magnetic phenomena.
✘ Electromagnetics is the study of the effects of electric charges at rest
and in motion.
✘ Both positive and negative charges are sources of an electric field.
✘ Moving Charges produce a current, which gives rise to a magnetic field.
✘ A Field is a spatial distribution of a quantity, which may or may not be a
function of time. A time-varying electric field is accompanied by a
magnetic field, and vice versa resulting in electromagnetic field.
✘ Circuit theory Vs Field theory
✘ Inductive Approach Vs Deductive approach
✘ Electromagnetic model
✘ Units and Constants
2
UNIT-I: INTRODUCTION

Review of vector algebra
3
.
• Vector analysis is mathematical shorthand.
• Vector analysis is a mathematical tool with which
electromagnetic (EM) concepts are most conveniently expressed
and best comprehended.
• Three main topics on vector analysis are as follows:
1. Vector algebra – addition, subtraction, and multiplication of vectors.
2. Orthogonal coordinate systems – Cartesian, cylindrical, and spherical
Coordinates.
3. Vector calculus – differentiation and integration of vectors; line,
surface, and volume integrals;“del” operator; gradient, divergence,
and curl operations.

4
.
• Scalar: At a given time and position, a scalar is completely
specified by its magnitude. The term scalar refers to a quantity
whose value may be represented by a single (positive or
negative) real number.
• Examples:
• A body falling a distance L in a time t
• The temperature T at any point in a bowl of soup whose
coordinates are x, y, and z
Here L, t, T, x, y, and z are all scalars.
• Other scalar quantities are mass, density, pressure, volume,
charge, electric potential, voltage and population.

5
.
• Vector: The specification of a vector at a given
location and time requires both a magnitude
and a direction.
• Examples:
• Force, velocity, acceleration, displacement,
electric field intensity and a straight line from
the positive to the negative terminal of a
storage battery are examples of vectors.

6
.
• Field:
A field is a function that specifies a particular quantity
everywhere in a region. The value of a field varies in general
with both position and time.
• Both scalar fields and vector fields exist.
• The temperature throughout the bowl of soup, temperature
distribution in a building, sound intensity in a theater are
examples of scalar fields.
• The gravitational force on a body in space, the velocity of
raindrops in the atmosphere are examples of vector fields.

Review of vector algebra 7
.
• Unit Vector: A vector A can be written as
 A is the magnitude of A.
The vector A can be represented graphically by a directed straight-line segment
of a length |A| = A with its arrowhead pointing in the direction of aA
 aA is a unit vector with a unity magnitude having the direction of A.

8
.
 A vector A in Cartesian (or rectangular) coordinates may be represented as
where Ax, Ay, and Az are called the components of A in the x, y, and z directions respectively;
ax, ay and az are unit vectors in the x, y, and z directions, respectively.
The magnitude
of vector A is
given by
The unit vector along A is given by

.
• Position Vector:
 A point P in Cartesian coordinates may be represented by (x, y, z).
 The position vector rp (or radius vector) of point P is defined as the
directed distance from the origin O to P given by

• Distance vector (or separation vector):
 It is the displacement from one point to another.
 If two points P and Q are given by (xP, yP, zP) and (xQ, yQ, zQ), the
distance vector (or separation vector) is the displacement from P to Q.

SOLVED PROBLEMS-vector algebra 11

Specify the unit vector extending from the origin toward the point
G( 2, –2, –1).
Solution:

• Vector Addition:
 Two vectors A and B, which are not in the same direction nor in opposite
directions, determine a plane. Their sum is another vector C in the same plane.
 Vector addition, C = A + B can be obtained graphically in two ways.
By the parallelogram rule: The resultant C is the diagonal vector of the
parallelogram formed by A and B drawn from the same point.
By the head-to-tail rule: The head of A connects to the tail of B. Their sum C
is the vector drawn from the tail of A to the head of B; and vectors A, B, and C
form a triangle

• Vector Addition obeys the commutative and associative laws
 Commutative law: A + B = B + A
 Associative law: A + (B + C) = (A + B) + C
 - B is the negative of vector B; that is, -B has the same magnitude as B, but
its direction is opposite to that of B.
• Vector Subtraction can be defined in terms of vector addition in the
following way: A-B = A+(-B)

• Multiplication of a Vector by a Scalar:
 Multiplication of a vector A by a positive scalar k changes the magnitude of A
by k times without changing its direction.
 The magnitude of the vector changes, but its direction does not when the scalar is
positive, although it reverses direction when multiplied by a negative scalar.
 Multiplication of a vector by a scalar also obeys the associative and
distributive laws of algebra

• 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
• Scalar or Dot Product
 The scalar or dot product of two vectors A and
B, denoted by A • B, is a scalar, which equals
the product of the magnitudes of A and B and
the cosine of the angle between them.
θAB is the smaller angle between A and B and is less than π radians (180°).

 If A =(Ax, Ay, Az) and B = (Bx, By, Bz), then
A . B = AxBx + AyBy + AzBz
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
Hence A • B =0

Dot product obeys the following:
(i) Commutative law:
A . B = B . A
(ii) Distributive law:
A . (B + C) = A . B + A . C
(iii) Also note that
A . A = |A|2
ax . ay = ay . az = az . ax = 0
ax . ax = ay . ay = az . az = 1
Application:
ax . ay =1.1.Cos90=1.1.0=0
ax . ax =1.1.Cos0= 1.1.1=1
In mechanics, when a constant force F applied over a straight displacement L does
an amount of work FL cos θ,which is more easily written F · L.

Problems on Scalar or Dot Product
20
Dot Product A . B = AxBx + AyBy + AzBz

 The cross product of two vectors A and B, denoted by A x B, is
a vector.
• Cross Product or Vector Product
 The magnitude of A x B is equal to the product of the
magnitudes of A and B
 The sine of the smaller angle between A and B
 The direction of A x B is perpendicular to the plane
containing A and B and is along one of the two possible
perpendiculars which is in the direction of advance of a
right-handed screw as A is turned into B.

 If A =(Ax, Ay, Az) and B = (Bx, By, Bz), then
• Cross Product or Vector Product
This is obtained by “crossing” terms in cyclic permutation, hence the name
cross product

• The Cross Product has the following basic properties:
(i) It is not commutative:
Reversing the order of the vectors A and B results in a unit vector
in the opposite direction.
(ii) It is not associative:
(iii) It is distributive:
(iv)
(v)

Problems on Vector Product
25
Unit vector normal to the plane:

ORTHOGONAL COORDINATE SYSTEMS
 The physical quantities that are being dealt in Electromagnetics are
functions of space and time.
 In a three-dimensiona1 space, a point can be located as the intersection
of three surfaces.
 When these three surfaces are mutually perpendicular to one another,
then it is known as an orthogonal coordinate system.
 An orthogonal system is one in which the coordinates are mutually
perpendicular.
 The three orthogonal coordinate systems that are most common and
useful are:
1. Cartesian (or rectangular) coordinate system
2. Cylindrical coordinate system
3. Spherical coordinate system
26

Cartesian (or Rectangular) Coordinate System (x,y,z)
✘ A point P(x1, y1, z1) in Cartesian coordinates is the intersection
of three planes specified by x = x1, y = y1 and z = z1
27

Cartesian (or Rectangular) Coordinate System (x,y,z)
28
The unit vectors of the rectangular coordinate system have unit
magnitude and are directed toward increasing values of their
respective variables.

Electromagnetic fields: Review of vector algebra

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Electromagnetic fields: Review of vector algebra