Understanding Cartesian Coordinate System in Electromagnetism, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

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EEE 224 ELECTROMAGNETIC THEORY

Cartesian

Coordinate System

METU -

NCC

Assist. Prof. Dr.

Özlem Özgün

Office

:^

S-

Phone

:^

661 2972

E-mail

:^

ozozgun@metu.edu.tr

Web

:^

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

2

Coordinate Systems

The physical quantities in EM (e.g.,

electric and magnetic fields) are vector fields

, which are functions of both space

and time. ™

To describe the variations of these

quantities in space, we need a suitablecoordinate system. ™

Laws of electromagnetism are valid in any

coordinate system. Coordinate system = frame of reference

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Orthogonal Coordinate Systems

3 PRIMARY “ORTHOGONAL”

COORDINATE SYSTEMS:

•^

CARTESIAN

•^

CYLINDRICAL

•^

SPHERICAL

Choice is based on^ the symmetry of

the problem

Examples: Sheets - CARTESIAN Wires/Cables - CYLINDRICAL Spheres - SPHERICAL

How do we choose a suitable coordinate system?

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Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates

P (x, y, z)^ P (R,

θ

,^ φ

)

P (r,

φ

, z)

x

y

z

P(x,y,z) Φ z

r

x^

y

z

P(r,

Φ

, z)

θ Φ

R z

y

x

P(R,

θ

,^ Φ )

Useful in problemshaving cylindricalsymmetry Useful in problemshaving sphericalsymmetry Useful in problemshaving rectangularsymmetry

r

Orthogonal Coordinate Systems

Alternative notation:

ρ

is used for r.

Alternative notation: r is used for R.

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Cartesian Coordinate System

8

Cartesian Coordinate System

10

Cartesian Coordinate System

You said earlier that vector quantities (either discreteor field)

have both

magnitude

and

direction.

But how do we specify direction

in 3-D space?

Do we use

coordinate values

(e.g., x, y,

z )??

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Cartesian Coordinate System

It is very important that youunderstand that

coordinates

only allow us to specifyposition

in 3-D

space.

They cannot be

used to

specify direction! The most convenient way tospecify the direction

of a

vector quantity is by using awell-defined orthonormal

set

of

vectors known as base (unit)

vectors.

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Properties:

1. Each vector is a unit

vector:

⋅^

=

⋅^

=

⋅^

=

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

0

x^

y^

x^

z^

y^

z

a

a

a

a

a

a

⋅^

=

⋅^

=

⋅^

=

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

1

x^

x^

y^

y^

z^

z

a

a

a

a

a

a

2. Vectors are

mutually orthogonal:

(magnitude)

Cartesian Base (Unit)

Vectors

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Properties: (cont..)

3. Unit vectors must be arranged such that:

×

=

×

=

×

=

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

x^

y^

z

y^

z^

x

z^

x^

y

a

a

a

a

a

a

a

a

a

An orthonormal

set with this property

is known as a

right-handed

system.

Cyclic relation

All base vectors must form a right-handed orthonormal set.

Cartesian Base (Unit)

Vectors

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Cartesian Coordinate System

Any vector

can be expressed

in terms of the unit vectors.

=

ˆ

ˆ

ˆ

x^

x

y^

y^

z^

z

A

A a

A a

A a

ˆ x x

A a

ˆ y y

A a

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Cartesian Coordinate System

How to determine the scalar components A

, Ax

, Ay

z^

Use dot product!

But, since the unit vectors are

orthogonal,

we know that:

=

⋅

=

⋅

=

⋅

ˆ ˆ ˆ

x

x

y

y

z

z

A

A

a

A

A

a

A

A

a

Similarly,

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Cartesian Coordinate System

Expressing a vector in termsof 3 base vectors

makes life

easier. The

evaluation of vector operations (such as addition, subtraction, multiplication, dotproduct, and cross

product)

all

become straightforward if all vectors are

expressed

using the same set of base vectors.

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Cartesian Coordinate System

How can we find:

+^

−^

⋅^

×^

A

B

A

B

A

B A

B

etc