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CHAPTER 1

ELECTROMAGNETIC FIELDS THEORY

Vector Fields

Scalar

- Scalar A quantity that has only magnitude.
- For example time, mass, distance, temperature and

population are scalars. - Scalar is represented by a letter e.g., A, B

Vector

- Vector A quantity that has both magnitude and

direction. - Example Velocity, force, displacement and

electric field intensity. - Vector is represent by a letter such as A, B,

or - It can also be written as
- where A is which is the magnitude and

is unit vector

Unit Vector

- A unit vector along A is defined as a vector

whose magnitude is unity (i.e., 1) and its

direction is along A. - It can be written as aA or
- Thus

Vector Addition

- The sum of two vectors for example vectors A and

B can be obtain by moving one of them so that its

terminal point (tip) coincides with the initial

point (tail) of the other

Terminal point

initial

initial

Terminal point

Vector Subtraction

- Vector subtraction is similarly carried out as

- D A B A (-B)

Figure (c)

Figure (a)

Figure (c) shows that vector D is a vector that

is must be added to B to give vector A So if

vector A and B are placed tail to tail then

vector D is a vector that runs from the tip of B

to A.

Figure (b)

Vector multiplication

- Scalar (dot ) product (AB)
- Vector (cross) product (A X B)
- Scalar triple product A (B X C)
- Vector triple product A X (B X C)

Multiplication of a vector by a scalar

- Multiplication of a scalar k to a vector A gives

a vector that points in the same direction as A

and magnitude equal to kA - The division of a vector by a scalar quantity is

a multiplication of the vector by the reciprocal

of the scalar quantity.

Scalar Product

- The dot product of two vectors and ,

written as is defined as the

product of the magnitude of and , and

the projection of onto (or vice versa).

- Thus
- Where ? is the angle between and . The

result of dot product is a scalar quantity.

Vector Product

- The cross (or vector) product of two vectors A

and B, written as is defined as - where a unit vector perpendicular to the

plane that contains the two vectors. The

direction of is taken as the direction of the

right thumb (using right-hand rule) - The product of cross product is a vector

Right-hand Rule

Components of a vector

- A direct application of vector product is in

determining the projection (or component) of a

vector in a given direction. The projection can

be scalar or vector. - Given a vector A, we define the scalar component

AB of A along vector B as - AB A cos ?AB AaB cos ?AB
- or AB AaB

Dot product

- If and then
- which is obtained by multiplying A and B

component by component. - It follows that modulus of a vector is

Cross Product

- If A(Ax, Ay, Az), B(Bx, By, Bz) then

Cross Product

- Cross product of the unit vectors yield

Example 1

- Given three vectors P
- Q
- R
- Determine
- (PQ) X (P-Q)
- Q(R X P)
- P(Q X R)
- P X( Q X R)
- A unit vector perpendicular to both Q and R

Solution

Solution (cont)

Solution (cont)

- To find the determinant of a 3 X 3 matrix, we

repeat the first two rows and cross multiply

when the cross multiplication is from right to

left, the result should be negated as shown

below. This technique of finding a determinant

applies only to a 3 X 3 matrix. Hence

Solution (cont)

Solution (cont)

Solution (cont)

Cylindrical Coordinates

- Very convenient when dealing with problems having

cylindrical symmetry. - A point P in cylindrical coordinates is

represented as (?, F, z) where - ? is the radius of the cylinder radial

displacement from the z-axis - F azimuthal angle or the angular displacement

from x-axis - z vertical displacement z from the origin (as

in the cartesian system).

Cylindrical Coordinates

Cylindrical Coordinates

- The range of the variables are
- 0 ? lt 8, 0 F lt 2p , -8 lt z lt 8
- vector in cylindrical coordinates can be

written as (A?,Af, Az) or A?a? Afaf Azaz - The magnitude of is

Relationships Between Variables

- The relationships between the variables (x,y,z)

of the Cartesian coordinate system and the

cylindrical system (?, f , z) are obtained as - So a point P (3, 4, 5) in Cartesian coordinate is

the same as?

Relationships Between Variables

- So a point P (3, 4, 5) in Cartesian coordinate is

the same as P ( 5, 0.927,5) in cylindrical

coordinate)

Spherical Coordinates (r,?,f)

- The spherical coordinate system is used dealing

with problems having a degree of spherical

symmetry. - Point P represented as (r,?,f) where
- r the distance from the origin,
- ? called the colatitude is the angle between

z-axis and vector of P, - F azimuthal angle or the angular displacement

from x-axis (the same azimuthal angle in

cylindrical coordinates).

Spherical Coordinates

Spherical Coordinates (r,?,f)

- The range of the variables are
- 0 r lt 8, 0 ? lt p , 0 lt f lt 2p
- A vector A in spherical coordinates written as
- (Ar,A?,Af) or Arar A?a? Afaf
- The magnitude of A is

Relation to Cartesian coordinates system

Relationship between cylinder and spherical

coordinate system

Point transformation

Point transformation between cylinder and

spherical coordinate is given by or

Example

- Express vector B
- in Cartesian and cylindrical coordinates. Find B

at (-3, 4 0) and at (5, p/2, -2)

Differential Elements

- In vector calculus the differential elements in

length, area and volume are useful. - They are defined in the Cartesian, cylindrical

and spherical coordinate

Cartesian Coordinates

Differential elements

dy

z

P Q S R

dz

B

A

dx

D C

az

y

y

ax

ay

x

x

Differential displacement

Cartesian Coordinates

Differential elements

Differential normal area

Differential elements

Cartesian Coordinates

Cylindrical Coordinates

Differential elements

Differential displacement

Cylindrical Coordinates

Differential elements

Differential normal area

Cylindrical Coordinates

Differential elements

Spherical Coordinates

Differential elements

Differential displacement

Spherical Coordinates

Differential elements

Differential normal area

Spherical Coordinates

Differential elements

Del Operator

- Written as is the vector differential

operator. Also known as the gradient operator.

The operator in useful in defining

1. The gradient of a scalar V, written as

V 2. The divergence of a vector A, written as 3.

The curl of a vector A, written as 4. The

Laplacian of a scalar V, written as

Gradient of Scalar

- G is the gradient of V. Thus
- In cylindrical coordinates,
- In spherical coordinates,

Example

Divergence

- In Cartesian coordinates,
- In cylindrical coordinates,
- In spherical coordinate,

Example

Curl of a Vector

- In Cartesian coordinates,
- In cylindrical coordinates,

Curl of a Vector

- In spherical coordinates,

Examples on Curl Calculation

Laplacian of a scalar

- The Laplacian of a scalar field V, written as ?2V

is defined as the divergence of the gradient of

V. - In Cartesian coordinates,

Laplacian of a scalar

- In cylindrical coordinates,
- In spherical coordinates,

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