Introduction to Electromagnetism Orthogonal Coordinate Systems
(2-4, 2-5)
Yoonchan Jeong
School of Electrical Engineering, Seoul National University Tel: +82 (0)2 880 1623, Fax: +82 (0)2 873 9953
Email: [email protected]
2
Orthogonal Coordinate systems (1)
Orthogonal coordinate system:
How many surfaces are required for determining the location of a point in a three-dimensional space?
A point can be defined as the intersection of three surfaces.
If these three surfaces are mutually perpendicular to one another Base vectors:
The unit vectors perpendicular to the three surfaces
=
×
=
×
=
×
→
2 1
3
1 3
2
3 2
1
u u
u
u u
u
u u
u
a a
a
a a
a
a a
a
1 3
3 2
2
1 u u u u u
u a a a a a
a ⋅ = ⋅ = ⋅
→
3 3
2 2
1
1 u u u u u
u a a a a a
a ⋅ = ⋅ = ⋅
→ Vector representation:
3 3 2
2 1
1 u u u u u
u A a A a A
a
A = + +
→
2 / 1 2 2
2 )
( A u 1 A u 2 A u 3
A = = + +
→ A
Right-handed system
= 0
= 1
3
Orthogonal Coordinate systems (2)
Vector product:
Example 2-4:
Scalar product:
3 3 2
2 1
1
3 3 2
2 1
1
u u u
u u
u
u u u
u u
u
B B
B
A A
A
a a
a B
a a
a A
+ +
=
+ +
=
) (
)
( a u 1 A u 1 a u 2 A u 2 a u 3 A u 3 a u 1 B u 1 a u 2 B u 2 a u 3 B u 3
B
A ⋅ = + + ⋅ + +
3 3 2
2 1
1 u u u u u
u B A B A B
A + +
=
=
×
=
×
=
×
→
2 1
3
1 3
2
3 2
1
u u
u
u u
u
u u
u
a a
a
a a
a
a a
a 0
1 3
3 2
2
1 ⋅ = ⋅ = ⋅ =
→ a u a u a u a u a u a u
3 1
3 2
2 1
1 ⋅ = ⋅ = ⋅ =
→ a u a u a u a u a u a u Recall:
) (
)
( a u 1 A u 1 a u 2 A u 2 a u 3 A u 3 a u 1 B u 1 a u 2 B u 2 a u 3 B u 3
B
A × = + + × + +
3 2
1
3 2
1
3 2
1
u u
u
u u
u
u u
u
B B
B
A A
A
a a
a
=
) (
) (
)
( 2 3 3 2 2 3 1 1 3 3 1 2 2 1
1 u u u u u u u u u u u u u u
u A B − A B + A B − A B + A B − A B
= a a a
4
Conversion factors
Some of the coordinates, say u i (i = 1, 2, or 3), may not be a length:
A conversion factor is needed to convert a differential change du i into a change in length dl i :
i i
i h du
dl = ← h i is called a “ metric coefficient”
Directed differential length change: 1 2 3
3 2
1 dl dl dl
d l = a u + a u + a u
) (
) (
)
( 1 1 2 2 3 3
3 2
1 h du h du h du
d l = a u + a u + a u
→
Differential volume change: dv = dl 1 dl 2 dl 3
3 2 1 3 2
1 h h du du du h
dv =
→
Differential area change: d s = a n ds → ds 1 = dl 2 dl 3
3 2 3 2
1 h h du du
ds =
→
3 1 3 1
2 h h du du
ds =
→
2 1 2 1
3 h h du du
ds =
→
5
Cartesian (or Rectangular) Coordinates (1)
Notation: ( u 1 , u 2 , u 3 ) = ( x , y , z )
Vector representation:
=
×
=
×
=
×
→
y x
z
x z
y
z y
x
a a
a
a a
a
a a
a
1 1
1 y z
x
OP = a x + a y + a z
→
z z y
y x
x A a A a A
a
A = + +
→ Dot product:
y y y
y x
x B A B A B
A + +
=
⋅
→ A B Cross product:
) (
) (
)
( y z z y y z x x z z x y y x
x A B − A B + A B − A B + A B − A B
=
×
→ A B a a a
z y
x
z y
x
z y
x
B B
B
A A
A
a a
a
=
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
6
Directed differential length change: d l = a x dx + a y dy + a z dz
Differential volume change: dv = dxdydz Differential area change:
dxdy ds
dxdz ds
dydz ds
z y x
=
=
=
Cartesian (or Rectangular) Coordinates (2)
Metric coefficients:
3 1
2
1 = = =
→ h h h
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
7
Cylindrical Coordinates (1)
Notation: ( u 1 , u 2 , u 3 ) = ( r , φ , z )
Vector representation:
=
×
=
×
=
×
→
φ φ
φ
a a
a
a a
a
a a
a
r z
r z
z r
z z r
r A a A a A
a
A = + +
→ φ φ
Dot product:
z z r
r B A B A B
A + +
=
⋅
→ A B φ φ
Cross product:
) (
) (
)
( z z z r r z z r r
r A φ B − A B φ + φ A B − A B + A B φ − A φ B
=
×
→ A B a a a
z r
z r
z r
B B
B
A A
A
φ φ
φ a
a a
=
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
8
Directed differential length change: d l = a r dr + a φ rd φ + a z dz
Cylindrical Coordinates (2)
Metric coefficients:
3 1
1 = =
→ h h r h =
→ 2
Differential volume change: dv = rdrd φ dz Differential area change: ds r = rd φ dz
drdz ds φ =
φ rdrd ds z =
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
9
Cylindrical Coordinates (3)
Transformation from cylindrical coordinates to Cartesian coordinates:
z z r
r A a A a A
a
A = + φ φ +
x
A x = A ⋅ a
→
x x
r
r A
A a ⋅ a + a ⋅ a
= φ φ
φ φ φ sin
cos A
A r −
=
y
A y = A ⋅ a
→
y y
r
r A
A a ⋅ a + a ⋅ a
= φ φ
φ φ φ cos
sin A
A r +
=
z
z A
A =
→
−
=
→
z r
z y x
A A A
A A A
φ φ
φ
φ φ
1 0
0
0 cos
sin
0 sin
cos
z z
r y
r x
=
=
=
φ φ sin cos
z z
x y y x
r
=
=
+
=
−1 2 2
φ tan Conversion formulas:
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
10
Spherical Coordinates (1)
Notation: ( u 1 , u 2 , u 3 ) = ( R , θ , φ )
Vector representation:
=
×
=
×
=
×
→
θ φ
φ θ
φ θ
a a
a
a a
a
a a
a
R
R R
φ φ θ
θ A A
A R
R a a
a
A = + +
→ Dot product:
φ φ θ
θ B A B
A B
A R R + +
=
⋅
→ A B Cross product:
) (
) (
)
( R R R R
R A θ B φ − A φ B θ + θ A φ B − A B φ + φ A B θ − A θ B
=
×
→ A B a a a
φ θ
φ θ
φ θ
B B
B
A A
A
R R
R a a
a
=
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
11
Directed differential length change: d l = a R dR + a θ Rd θ + a φ R sin θ d φ
Spherical Coordinates (2)
Metric coefficients:
1 = 1
→ h
R h =
→ 2
Differential volume change: dv = R 2 sin θ dRd θ d φ Differential area change: ds R = R 2 sin θ d θ d φ
θ
3 R sin h =
→
φ
θ R θ dRd
ds = sin
φ RdRd θ
ds =
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
12
Spherical Coordinates (3)
Transformation from spherical coordinates to Cartesian coordinates:
θ
φ θ
φ θ
cos
sin sin
cos sin
R z
R y
R x
=
=
=
x y
z y x
z y
x R
1
2 2
1
2 2
2
tan tan
−
−
=
= +
+ +
=
φ θ Conversion formulas:
See Example 2-11 (HW)
13
Three Basic Orthogonal Coordinate Systems
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
14
Integrals Containing Vector Functions
Common forms of integrals containing vector functions:
V dv
∫ F
l d
C V
∫
l
C F d ⋅
∫
s
S A d ⋅
∫
Defining the surface normal vector:
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
Surface normal vectors?
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
