Page 20 - Engineering Electromagnetics, 8th Edition

2 ENGINEERING ELECTROMAGNETICS

n-dimensional space in more advanced applications. Force, velocity, acceleration,
and a straight line from the positive to the negative terminal of a storage battery
are examples of vectors. Each quantity is characterized by both a magnitude and a
direction.
Our work will mainly concern scalar and vector fields.A field (scalar or vector)
may be defined mathematically as some function that connects an arbitrary origin
to a general point in space. We usually associate some physical effect with a field,
such as the force on a compass needle in the earth’s magnetic field, or the movement
of smoke particles in the field defined by the vector velocity of air in some region
of space. Note that the field concept invariably is related to a region. Some quantity
is defined at every point in a region. Both scalar fields and vector fields exist. The
temperature throughout the bowl of soup and the density at any point in the earth
are examples of scalar fields. The gravitational and magnetic fields of the earth, the
voltage gradient in a cable, and the temperature gradient in a soldering-iron tip are
examples of vector fields. The value of a field varies in general with both position and
time.
In this book, as in most others using vector notation, vectors will be indicated by
boldface type, for example, A. Scalars are printed in italic type, for example, A. When
writing longhand, it is customary to draw a line or an arrow over a vector quantity to
show its vector character. (CAUTION: This is the first pitfall. Sloppy notation, such as
the omission of the line or arrow symbol for a vector, is the major cause of errors in
vector analysis.)

1.2 VECTOR ALGEBRA

With the definition of vectors and vector fields now established, we may proceed to
define the rules of vector arithmetic, vector algebra, and (later) vector calculus. Some
of the rules will be similar to those of scalar algebra, some will differ slightly, and
some will be entirely new.
To begin, the addition of vectors follows the parallelogram law. Figure 1.1 shows
the sum of two vectors, A and B.Itis easily seen that A + B = B + A,or that vector
addition obeys the commutative law. Vector addition also obeys the associative law,

A + (B + C) = (A + B) + C
Note that when a vector is drawn as an arrow of finite length, its location is
defined to be at the tail end of the arrow.
Coplanar vectors are vectors lying in a common plane, such as those shown
in Figure 1.1. Both lie in the plane of the paper and may be added by expressing
each vector in terms of “horizontal” and “vertical” components and then adding the
corresponding components.
Vectors in three dimensions may likewise be added by expressing the vectors
in terms of three components and adding the corresponding components. Examples
of this process of addition will be given after vector components are discussed in
Section 1.4.