EEE 224 ELECTROMAGNETIC THEORY: Cylindrical Coordinate System, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

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EEE 224 ELECTROMAGNETIC THEORY^ Cylindrical 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 Cartesian Coordinates^ P (x, y, z) Cylindrical Coordinates Spherical Coordinates^ P (R,^ θ, P (r,^ φ, z)^ φ)

x z^ P(x,y,z)^ y^ z^ r^ Φ x^ z^ P(r,^ Φ, z) y^ z^ θ^ R^ y Φ P(R, x Useful in problemshaving rectangularsymmetry r^ Useful in problemshaving cylindricalsymmetry θ, Φ)^ Useful in problemshaving sphericalsymmetry

Orthogonal Coordinate Systems Alternative notation:^ ρ^ is used for r. Alternative notation: r is used for R.

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Cylindrical Coordinate System^ r

φ( , )P r

r

φ( , ,^ )P r z 5

Cylindrical Coordinate System

7 r-const surface: surface of circular cylinder (red surface) φ -const surface: semi-infinite plane

(yellow surface) z-const surface: horizontal

infinite plane

(blue surface)

™^ A point can be specified as the intersection of 3 surfaces.^ r^ = constant^ φ^ = constant^ z^ = constant

z P x y

Cylindrical Coordinate System

8 r-const surface: surface of circular cylinder φ -const surface: semi-infinite plane z-const surface: horizontal

infinite plane

r^ = const.

z^ = const. φ^ = const.

Cylindrical Coordinate System

10 EXERCISE: Define this green region using coordinate variables!

φ =^ φ^1 z= z^2 z= z^1 φ =^ φ^2

r= r^1 r= r^2

≤^ ≤ r r^ r (^1 2) φ φ^ φ≤^ ≤ 1 2 ≤^ < z z^ z 1 2

Cylindrical Coordinate System

11 EXERCISE: Define this region using coordinate variables!^ φ =^ φ^1

φ =^ φ^2 z= z^1 r= r^1 z= 0

0 ≤^ ≤^ r^ r^1 φ^ φ^ φ≤^ ≤^1 20 ≤^ < z^ z^1

Cylindrical Coordinate System

13 : always points in the direction of increasing

r, and is orthogonal to the

r-constant plane (cylindrical surface). (It points away from the z-axis.)^ : always points in the direction of increasing

φ , and is orthogonal to the

φ^ -constant plane. It is tangent to the cylindrical surface.^ : always points in the direction of increasing

z, and is orthogonal to the

z-constant plane.

Cylindrical Base (Unit) Vectors r a^ r

ˆar ˆaφ^ ˆaz

Properties: 1. Each vector is a unit 14

vector: ˆ^ ˆ^ ˆ^

ˆ^ ˆ^ ˆ^

0 rr^

zz a^ a^ a^

a^ a^ a⋅ = ⋅ =^ ⋅^ φφ

= ˆ^ ˆ^ ˆ^

ˆ^ ˆ^ ˆ^

1 r^ r

z^ z a^ a^ a^

a^ a^ a⋅ = ⋅ =^ ⋅^ φ φ

=

2. Vectors are (magnitude)^ mutually orthogonal:

Cylindrical Base (Unit) Vectors^ r

a^ r

3. Unit vectors must be arranged

such that:^ Cyclic relation ˆ^ ˆ^ ˆa^ a^ a×^ =rz^ φ ˆ^ ˆ^ ˆa^ a^ a×^ =z^ rφ^ ˆ^ ˆ^ ˆa^ a^ a×^ =z^ rφ

Cylindrical base vectors are dependent on position.

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Transformation between Cylindrical and Cartesian Coordinates^ The relationship between a

,a, aand ax yz^

The relationship between(x,y,z) and (r,^ φ , z)^ cosx^ r^ φ^ = siny^ r^ φ^ = z^ z=^ ,a, ar^ φ z

2 2 r x y= +^1 −tan /y^ x φ =(^ ) z z=

(^ )^ (

)^ (^

(^ )^ (

(^ )

ˆ^ ˆ^ ˆ^ ˆ

ˆ^ ˆ^ ˆ^

ˆ^ ˆ^ ˆ^

ˆ^ ˆcos sin

ˆ^ ˆ^ ˆ^ ˆ

ˆ^ ˆ^ ˆ^

ˆ^ ˆ^ ˆ^

ˆ^ ˆsin cos

rr^ x^ ˆ^ ˆ xr^ y^ y

r^ z^ z

xy

x^ xy

yz^

zx

y

a^ a^ a^ a^ z^ z

a^ a^ a^ a

a^ a

aa

a^ a^ a^ a

a^ a^ a^

a^ a^ a

aa

φφ a^ a

φφ

=^ ⋅^ +

⋅^ +^

⋅^ =^

=^ ⋅^ +

⋅^ +^

⋅^ = −^

ˆ^ ˆ^ =

ˆ^ ˆ^

ˆ^ ˆ^

ˆ^ ˆ

cos^

sin^

sin^

cos

r^ x

r^ y

x

y

a^ a

a^ a

a^ a

a^ a

φ

φ

⋅^ =^

⋅^ =^

⋅^ = −^

⋅^ =

ˆ ax x

y^ ˆ ay 0

ˆ ar

ˆ a^ φ φ^ φ y^ P^ r^ φ^ x

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Transformation between Cylindrical and Cartesian Coordinates^ The relationship between A

,A, Aand Ax yz^

,A, Ar φ z

cos^ sin^

0 sin^ cos^

0 0 0

1 x

r y z

z A

A A

A A

φ A φ^ φ−⎡ ⎤ ⎡ φ^ φ

⎤ ⎡^ ⎤ ⎢^ ⎥^ ⎢^

⎥ ⎢^ ⎥ =⎢ ⎥ ⎢^

⎥ ⎢^ ⎥ ⎢^ ⎥^ ⎢^

⎥ ⎢^ ⎥ ⎣^ ⎦^ ⎣^

ˆ^ ˆ^ ⎦ ⎣ ⎦

ˆ^ ˆ^

x^ x^ y^ y^

z^ z r^ r

z^ z A^ A a^ A a

A a A^ A a^ Aa =^ +^ +^ A a=^ +^ +^ φ^ φ^ cos

sin^0 r sin^ cos^00 0

x y z

z A

A A

A A

A φ^ φ φ φ φ ⎡^ ⎤^ ⎡^

⎤ ⎡^ ⎤ ⎢^ ⎥^ ⎢^

⎥ ⎢^ ⎥ =^ −⎢ ⎥ ⎢^

⎥ ⎢^ ⎥ ⎢^ ⎥^ ⎢^

⎥ ⎢^ ⎥ ⎣^ ⎦^ ⎣^

(^ ) (^ ) ⎦ ⎣^ ⎦ ˆ^ ˆ^

ˆ^ ˆ^ ˆ

ˆ^ ˆ^

ˆ^ ˆ^ ˆ

r^ rr^ x^ ˆ

x^ y^ y^ z^ z x x^ y^ y^ z^ z

A^ a^ A^ a^ z^ zz

A a^ A a^ A a

A^ a^ A^ a^

A a^ A a^ A a

=^ ⋅^ =^ φ φφ A a^ A^ A

⋅^ +^ + = ⋅ = ⋅^ +^ + = ⋅ =

(^

) (^

) ˆ^ ˆ^

ˆ^ ˆ^ ˆ

ˆ^ ˆ^

ˆ^ ˆ^ ˆ

x^ xx^ r ˆ

rz^

z y^ yy^

r^ rz^

z

A^ a^ A^ a^ z^ zz

A a^ A a^ A a

A^ a^ A^ a

A a^ A a^

A a

A^ a^ A^ A

φ^ φ φ^ φ =^ ⋅^ =^ ⋅^

+^ +

=^ ⋅^ =^ ⋅^

+^ +

=^ ⋅^ =