Note for Applied Physics-II – Modern Physics - AP-II By Suhas Mondal

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Module 1: A Crash Course in Vectors Lecture 1: Scalar and Vector Fields Objectives 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 etc. 1

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Elementary Vector Algebra: Geometrically a vector is represented by a directed line segment. Since a vector remains unchanged if it is shifted parallel to itself, it does not have any position information. A three dimensional vector can be specified by an ordered set of three numbers, called its components. The magnitude of the components depends on the coordinate system used. In electromagnetism we usually use Cartesian, spherical or cylindrical coordinate systems. (Specifying a vector by its components has the advantage that one can extend easily to n dimensions. For our purpose, however, 3 dimensions would suffice.) A vector represented by vector is given by is in Cartesian (rectangular) coordinates. The magnitude of the A unit vector in any direction has a magnitude (length) 1. The unit vectors parallel to the cartesian and coordinates are usually designated by these unit vectors, the vector and respectively. In terms of is written 2

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Any vector in 3 dimension may be written in this fashion. The vectors are said to form a basis. In fact, any three non-collinear vectors may be used as a basis. The basis vectors used here are perpendicular to one another. A unit vector along the direction of is Vector Addition Sum of two vectors and is a third vector. If then Geometrically, the vector addition is represented by parallelogram law or the triangle law, illustrated below. Scalar Multiplication The effect of multiplying a vector by a real number is to multiply its magnitude by without a change in direction (except where is negative, in which case the vector gets inverted). In the component representation, each component gets multiplied by the scalar Scalar multiplication is distributive in addition, i.e. Two vectors may be multiplied to give either a scalar or a vector. 3

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Scalar Product (The Dot products) The dot product of two vectors and is a scalar given by the product of the magnitudes of the vectors times the cosine of the angle ( ) between the two In terms of the components of the vectors Note that Dot product is commutative and distributive Two vectors are orthogonal if Dot products of the cartesian basis vectors are as follows Exercise 1 Show that the vectors and are orthogonal. Vector Product (The Cross Product) The cross product of two vectors is a vector whose magnitude is where is the angle between the two vectors. The direction of the product vector is , 4