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EEE 224 ELECTROMAGNETIC THEORY

Spherical

METU -^ Coordinate Systems

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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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)^ z^ r^ Φ x

y z^ P(r,

Φ, z) z^ θ^ R Φ

y x

P(R,^ θ,^

Useful in problemshaving rectangularsymmetry^ Useful in problemshaving cylindricalsymmetry Φ)^ Useful in problemshaving sphericalsymmetry r

Orthogonal Coordinate Systems

Alternative notation:

ρ^ is used for r. Alternative notation: r is used for R.

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Spherical Coordinate System

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Spherical Coordinate System

R:^ distance from the origin (i.e., similar to altitude).

(0≤R<

θ :^ angle formed with the z-axis (i.e., similar to latitude) (

≤ θ ≤π).

φ :^ rotation angle around the z-axis

(i.e., similar to longitude).

(0≤ φ <

π). It is the same as the

coordinate

φ^ used in

Coordinate variables(u, u^1 cylindrical coords.

, u) = 23

(R,^

θ ,^ φ )

R^ Æ^ length θ^ ,^ φ^

Æ^ angle

R=3.

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R^ = const.

φ^ = const.

Spherical Coordinate System

θ^ = const.

R-const surface: spherical surface θ -const surface: cone surface φ -const surface: semi-infinite plane

MOVIE^8

Spherical Coordinate System

In this movie,consider r as R in our case. Also, interchange

θ^ and^

φ.

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Spherical Base (Unit) Vectors

ˆa^ : always points in the direction of increasing R ˆaθ ˆaφ

R, and is orthogonal to

theR-constant plane (spherical surface).^ : always points in the direction of increasing

θ , and is orthogonal to the

θ^ -constant plane

(cone surface). It is tangent to the sphere along the line of longitude.^ : always points in the direction of increasing

φ , and is orthogonal to the

φ^ -constant plane. It is tangent to the sphere along the line of latitude.

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Spherical Base (Unit) Vectors

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Vector

representation

in cylindrical coordinates:

ˆ^

ˆ^

R^ R

A^

A a^

A a^

A a

θ^ θ^

φ^ φ

=^

+^

Spherical Coordinate System

Be careful! To perform vector operations (e.g.,addition, dot product, position vector,etc), first convert sphericalcoordinates to cartesian coordinates.After doing the operations, transformthem back to the sphericalcoordinates. Reason:

θ^ and

φ^ are angles, not lengths.

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The relationship between (x,y,z),

(r,^ φ

, z) and (R,

θ ,^ φ ) sin^

cos sin^

sin cossin x^ R y^ R z^ R r^ R

θ^

φ θ^

φ θ θ = = = =^ (^

) 2 2

2 1 2 2 2

cos^1 tan^ /

R^

x^ y

z z x^ y

z y^ x θ φ

=^ +^ − −

⎜^

=^

⎜^

+^ +

⎝^

Transformation between Spherical, Cylindrical

and Cartesian Coordinates

x

y

P R z

r

cos z^ R θ=

sin r^ R^

θ

x

y sin

y^ r^

=

cos x^ r^

φ

16 ˆ ar r

z 0

ˆ az

ˆ aR ˆ a θ θ

P^ θ θ R

ˆ^ ˆ^

sin ˆ^ ˆ^

cos ˆ^ ˆ^

cos ˆ^ ˆ^ r sin a^ aR a^ azR a^ ar^ θ a^ azθ

θ θ θ θ ⋅^ = ⋅^ = ⋅^ = ⋅^ =^

−

ˆ ax x

y 0

ˆ ay

ˆ ar ˆ a^ φ^ φ

φ φ^ P^ x y^ r

ˆ^

ˆ^

ˆ cos^

sin ˆ^

ˆ^

ˆ sin^

cos x ˆ^ ˆ

r y

r a z^ z

a

a

a

a

a

a^ a

φ φ φ

φ φ

φ =^

− =^

=

Remember:

ˆ^ ˆ^

sin^ cos ˆ^ ˆ^

sin^ sin ˆ^ ˆ^

cos^ cos ˆ^ ˆ^ x cos^ sin

a^ aR a^ ayR a^ ax^ θ a^ ayθ

θ^ φ θ^ φ θ^ φ θ^ φ

⋅^ = ⋅^ = ⋅^ = ⋅^ =

Transformation between Spherical and Cartesian Coordinates

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Transformation between Spherical

and Cartesian Coordinates

The relationship between A

,A, Ax y

and Az

,A, AR θ

φ

sin^ cos^

cos^ cos

sin sin^ sin^

cos^ sin^

cos cos^

sin^

0

x^

R

A^ y z

A

A^

A

A^

θ A φ

θ^ φ^

θ^ φ^

φ

θ^ φ^

θ^ φ^

φ θ

θ

⎡^ ⎤−

⎡^ ⎤^ ⎡

⎤ ⎢^ ⎥

⎢^ ⎥^ ⎢

⎥

=^

⎢^ ⎥

⎢^ ⎥^ ⎢

⎥ ⎢^ ⎥

⎢^ ⎥^ ⎢

⎥ − ⎣^ ⎦^ ⎣

⎦ ⎣^ ⎦

sin^ cos sin^

sin^ cos cos^ cos

cos^

sin^ sin sin^

cos^

0

R^

x y z

A^

A

A^

A

A^

A

θ φ

θ^ φ^

θ^ φ^

θ θ^ φ^

θ^ φ^

θ φ

φ

⎡^ ⎤^ ⎡

⎤ ⎡^ ⎤

⎢^ ⎥^ ⎢

⎥ ⎢^ ⎥

=^

−

⎢^ ⎥^ ⎢

⎥ ⎢^ ⎥

⎢^ ⎥^ ⎢

⎥ ⎢^ ⎥

−⎣

⎦ ⎣^ ⎦

⎣^ ⎦

ˆ^

ˆ^

ˆ^

ˆ^

x^ x^

y^ y^

z^ z R^ R A^ A a

A a

A a A^ A a

A a

Aa θ θ^ φ^

φ

=^

+^

=^

+^

(^

) (^

) (^

)

ˆ^

ˆ^

ˆ^

ˆ^

ˆ^

ˆ^

ˆ^

ˆ^

ˆ^

ˆ^

ˆ^

ˆ^

r

r^ x^

x^ y^

y^ z^

z

R^

x^ x^

y^ y^

z^ z x^ x^

y^ y^

z^ z

A^ a^

A^ a^

A a^

A a^

A a

A^ a^

A^ a^

A a^

A a^

A a

A^ a^

A^ a^

A a^

A a^

A a

θ^ θ

θ φ^ φ

φ =^ ⋅^

=^ ⋅^

+^

=^ ⋅^

=^ ⋅^

+^

=^ ⋅^

=^ ⋅^

+^

(^

) (^

) (^

)

ˆ^

ˆ^

ˆ^

ˆ^ ˆ

ˆ^

ˆ^

ˆ^

ˆ^ ˆ

ˆ^

ˆ^

ˆ^

ˆ^ ˆ

x^ x

x^ R^

R

y^ y

y^ R^

R

z^ z

z^ R^

R

A^ a^

A^ a^

A a^

A a^ A a

A^ a^

A^ a^

A a^

A a^ A a

A^ a^

A^ a^

A a^

θ^ θ^ A a^ A a φ^ φ θ^ θ^ φ^ φ θ^ θ^ φ^ φ

=^ ⋅^

=^ ⋅^

+^

=^ ⋅^

=^ ⋅^

+^

=^ ⋅^

=^ ⋅^

+^

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Line Integral