# Scalar And Vector Fields

Physical quantities that we deal with in electromagnetism can be scalars or vectors fields. A scalar field is an entity that only has a magnitude for example mass, time, distance, electric charge, electric potential, energy, temperature, etc. A vector field is characterized by both magnitude and direction for example displacement, velocity, acceleration, force, electric field, magnetic field, etc. A quantity that can be defined everywhere in space as a function of the position is called field. The quantity that is specified may be scalar and vector fields.

## Vector Algebra

A three-dimensional vector can be defined by a set of three numbers known as its components. The magnitude of the components depends on the coordinate system used. In the electromagnetism system, we mostly 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 \vec { A } is represented by (Ax , Ay ,Az) in cartesian (rectangular) coordinates. The magnitude of the vector is given by

\left| A \right| =\sqrt { A^2_x+A^2_y+A ^2_z}

A unit vector in any direction has a magnitude 1.

The unit vectors parallel to the cartesian and coordinates x, y and z are usually designated by \hat { i } ,\hat { j }and\quad \hat{ k } respectively. In terms of these unit vectors, the vector is written

\overrightarrow { A } =\hat { i } A_x+\hat { j } A_y+\hat { k } A_z

The vectors \hat { i } ,\hat { j }and\quad \hat{ k } are form a basis. Any three non-colinear vectors may be used as a basis. The basis vectors used here are perpendicular to one another. A unit vector along the direction of \vec { A } is

\widehat { A } =\cfrac { \overrightarrow { A } }{ \left| A \right| }

Sum of two vectors \vec { A } and \vec { B } is a third vector.

\vec { A }=(Ax , Ay ,Az)

\vec { B }=(Bx , By ,Bz)

\vec { A }+\vec { B }=(Ax+Bx, Ay+By, Az+Bz)

The vector addition is represented by parallelogram law or the triangle law

## Scalar Multiplication

Multiplying a vector by a real number ‘c’ is to multiply its magnitude by ‘c’ without a change in direction. In the component representation, each component gets multiplied by the scalar

\vec { cA }=(cAx , cAy ,cAz)

Scalar multiplication is distributive in addition, i.e.

c(\vec { A }+\vec { B })=\vec { cA }+\vec { cB }

## Scalar Product (Dot Products)

The dot product of two vectors \vec { A } and \vec { B } is a scalar given by the product of the magnitudes of the vectors times the cosine of the angle ( \le 0\le \theta) between the two

\overrightarrow { A } \cdot \overrightarrow { B } =\left| A \right| \left| B \right| \cos { \theta }

In terms of the components of the vectors

\overrightarrow { A } \cdot \overrightarrow { B }=AxBx + AyBy +Az Bz

Dot product is commutative

\overrightarrow { A } \cdot \overrightarrow { B } =\overrightarrow { B } \cdot \overrightarrow { A }

Dot product is distributive

\overrightarrow { A } \cdot \overrightarrow { (B } +\overrightarrow { C) } =\overrightarrow { A } \cdot \overrightarrow { B } +\overrightarrow { A } \cdot \overrightarrow { C }

If two vectors are orthogonal then

\overrightarrow { A } \cdot \overrightarrow { B }=0

Dot products of the cartesian basis vectors are as follows

\hat { i } \cdot \hat { i } =\hat { j } \cdot \hat { j } =\hat { k } \cdot \hat { k } =1

\hat { i } \cdot \hat { j } =\hat { j } \cdot \hat { k } =\hat { k } \cdot \hat { i } =0

## Vector Product (Cross Product)

The cross product of two vectors \overrightarrow { A } \times \overrightarrow { B } is a vector whose magnitude is \left| A \right| \left| B \right| \sin { \theta } where \theta is the angle between the two vectors. The direction of the product vector is perpendicular to both \overrightarrow { A } and \overrightarrow { B }. The direction \overrightarrow { A } \times \overrightarrow { B } of is fixed by a convention, called the Right Hand Rule.

### Right Hand Rule

Put out the fingers of the right hand so that the thumb becomes perpendicular to both the forefinger and the middle finger. If the forefinger points in the direction of \vec { A } and the middle finger in the direction of \vec { B } then, \overrightarrow { A } \times \overrightarrow { B } points in the direction of the thumb. The rule is also called the Right-handed corkscrew rule which may be stated as follows.

If a right-handed screw is turned in the direction from \vec { A } to \vec { B } , the direction in which the head of the screw proceeds gives the direction of the cross product. In a cartesian basis, the cross product may be written in terms of the components of \vec { A } and \vec { A } as follows.

Vector product is anti-commutative

\overrightarrow { A } \times \overrightarrow { B }= –\overrightarrow { B } \times \overrightarrow { A }

Vector product is distributive

\overrightarrow { A } \times \left( \overrightarrow { B } +\overrightarrow { C } \right) =\overrightarrow { A } \times \overrightarrow { B } +\overrightarrow { A } \times \overrightarrow { C }

The cross product of cartesian basis vectors are

\hat { i } \times \hat { j } =\hat { k }

\hat { j } \times \hat { k } =\hat { i }

\hat { k } \times \hat { i} =\hat { j }

And

\hat { i } \times \hat { i } =\hat { j } \times \hat { j }=\hat { k } \times \hat { k }=0

## Scalar and Vector Triple Products

The scalar triple product of vectors \overrightarrow { A }, \overrightarrow { B } and \overrightarrow { C } is defined by

\overrightarrow { A } \cdot \left( \overrightarrow { B } \times \overrightarrow { C } \right) =\overrightarrow { B } \cdot \left( \overrightarrow { C } \times \overrightarrow { A } \right) =\overrightarrow { C } \cdot \left( \overrightarrow { A } \times \overrightarrow { B } \right)

In terms of the cartesian components, the product can be written in determinant form

\overrightarrow { B } \times \overrightarrow { C } gives the area of a parallelogram of sides \vec { B } and \vec { C } , the triple product \overrightarrow { A } \left( \overrightarrow { B } \times \overrightarrow { C } \right) gives the volume of a parallelopiped of sides \vec { A }, \vec { B } and \vec { C }.

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