Module 1: A Crash Course in Vectors
Lecture 1: Scalar and Vector Fields
In this lecture you will learn the following
Learn about the concept of field
Know the difference between a scalar field and a vector field.
Review your knowledge of vector algebra
Learn how an area can be looked upon as a vector
Define position vector and study its transformation properties under rotation.
SCALAR AND VECTOR FIELDS
This introductory chapter is a review of mathematical concepts required for the course. It is assumed
that the reader is already familiar with elementary vector analysis. Physical quantities that we deal with
in electromagnetism can be scalars or vectors.
A scalar is an entity which only has a magnitude. Examples of scalars are mass, time, distance, electric
charge, electric potential, energy, temperature etc.
A vector is characterized by both magnitude and direction. Examples of vectors in physics are
displacement, velocity, acceleration, force, electric field, magnetic field etc. A field is a quantity which
can be specified everywhere in space as a function of position. The quantity that is specified may be a
scalar or a vector. For instance, we can specify the temperature at every point in a room. The room may,
therefore, be said to be a region of ``temperature field" which is a scalar field because the temperature
is a scalar function of the position. An example of a scalar field in electromagnetism is the
electric potential. In a similar manner, a vector quantity which can be specified at every point in a
region of space is a vector field. For instance, every point on the earth may be considered to be in the
gravitational force field of the earth. we may specify the field by the magnitude and the direction of
acceleration due to gravity (i.e. force per unit mass )
at every point in space. As another
example consider flow of water in a pipe. At each point in the pipe, the water molecule has a velocity
. The water in the pipe may be said to be in a velocity field. There are several examples of
vector field in electromagnetism, e.g., the electric field
, the magnetic flux density