Vector Algebra: Scalars, Vectors, Addition, Multiplication, Fields, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

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EEE 224 ELECTROMAGNETIC THEORY

METU -^ Vector Algebra

NCC

Assist. Prof. Dr.

Özlem Özgün Office^

:^ S- Phone^

:^ 661 2972 E-mail^

:^ ozozgun@metu.edu.tr Web^

:^ http://www.metu.edu.tr/~ozozgun/

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Physical Quantities SCALAR VECTOR SCALAR FIELD VECTOR FIELD

DiscreteQuantities^ Distribution of aquantity

in space

Vector^4

A vector

is a quantity which has both a magnitude

and a

direction.

Aa

A^

ˆ= A

G G A A = Magnitude of the vector G A ˆ a =G A A

G A

Unit vector in the direction of

G A

G A

ˆ aA

G A A =

The length of the arrow is proportional to the magnitude

and

the orientation

indicates direction.

Vector^5

Examples

: velocity,

electric and magnetic field intensities. Wind velocity of a specific location at a specific time

10 m/s^

West

15 m/s^

North

Velocity

vector^ of a falling snow flake

Represented byboth magnitudeand direction

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Scalar Fields

Example

: Temperature

distribution in US

It cannot be specified with a single numeric value. Instead, we must specifytemperature as a function of position (location) on

the map.

e.g.: T(x,y) = 80 + 0.1x -0.2y + 0.0003xy + … where x and y are Cartesian coordinates that specify a point on the map.

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Vector Fields

Example: Surface wind velocity across U.S.

Vector field expression for wind velocity

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Vector Fields Example: Gravitational Field The gravitational field points toward tothe center of the Earth.

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Vector Fields

Example: Magnetic Field of a bar magnet

compass The needle aligns itself alongthe direction of magnetic field.

The Earth’s magnetic field behavesas if there were a bar magnet in it.

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VECTOR ALGEBRA^ • Vector Representations • Arithmetic Operations

of Vectors

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Vector Representations

REMEMBER:

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Vector Representations

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Vector Representations

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Vector Representations

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Arithmetic Operations

of Vectors • Addition • Subtraction • Multiplication (dot product, cross product)