2-4.3 Spherical Coordinates
EEE 340
Lecture 04
1
• A vector in spherical coordinates
 


A  aR AR  a A  a A
(2  65)
• The local base vectors from a right –handed
system



a R  a  a ,



a  a  a R ,



a  a R  a
EEE 340
Lecture 04
(2  64a)
(2  64b)
(2  64c)
2
The differential length
 


dl  aRdR  a Rd  a R sind
(2  66)
The differential areas are
2
dsR  R sin dd
ds  R sin dRd
(2  67)
ds  RdRd
The differential volume
dv  R sindRdd
2
EEE 340
Lecture 04
(2  68)
3
On many occasions the differential
areas are vectors

dsR  R sin  d d a R ;

ds  R sin  dR d a ;

ds  R dR d a
2
EEE 340
Lecture 04
4
Table 2-1 Basic Orthogonal Coordinates
(unit) base
vectors
differential
length
differential
areas
differential
volum e
EEE 340
Cartesian

a xor( xˆ )

a yor( yˆ )

a zor( zˆ )

dl  xˆdx  yˆ dy
 zˆdz
Cylindrical


aror(a  , ˆ , rˆ)

a or(ˆ)

a zor( zˆ )

dl  rˆdr  ˆrd
(2  44)  zˆdz
Spherical

aRor( Rˆ )

a or(ˆ)

a or(ˆ)

dl  Rˆ dR  ˆRd
(2  52)  ˆR sin d
(2  65)
(2  45)
(2  53)
(2  67)
dv  dxdydz
dv  rdrddz
dv  R 2 sin dRdd
(2  46)
(2  54)
Lecture 04
(2  68)
5
Cartesian coordinates



d  dx a x  dy a y  dz az

dS  dy dz a x

dx dz a y

dx dy a z
dv  dx dy dz
Differential
displacement
Differential
normal area
Differential
volume
d and dS are vectors.
dv is a scalar.
EEE 340
Lecture 04
6


dS  dx dy az  dS az
• The differential surface element dS may be
defined as dS  dS an
• we need to remember only d !
EEE 340
Lecture 04
7
Cylindrical coordinates



dS  r d dz ar ; dr dz a ; r d dr a z



d  dr ar  r d a  dz a z
dv   d d dz  rdrddz
EEE 340
Lecture 04
Differential
displacement
Differential
normal area
Differential
volume
8
Coordinate transforms
Example 2-11. Convert a vector in spherical
coordinates (SPC)




A  aR AR  a A  a A
into the Cartesian coordinates (CRT).
Solution. The general form of a vector in the CRT is



A  ax Ax  ay Ay  az Az

We need Ax  A  a x
In fact A  A a  a  A a  a  A a  a
x
R R
x
 
x
 
x
EEE 340
Lecture 04
9
 
aR  a x  sin  cos


x y
2
2
x y z
2
2
2

x y
2
2
(2  72)
x
x y z
2
x
2
2
The other eight dot-products can be worked out.
A faster and better way to represent the
transformation is based on the del operator.
EEE 340
Lecture 04
10
Example 2-12
Sphare chell
ra=2 cm
rb=5 cm
The charge density
 3 108
2
v 
cos 
4
R
C
m2
Find the total charge Q
EEE 340
Lecture 04
11
Q   ρdv 
• Solution:
2π π rb
   dφdθdrr
2
sinθρ
0 0 ra
2

rb
1
Q  3  10  cos d  sin d  2 dr
r
0
0
ra
8
2
 3  10  π  2  30
8
 1.8π  106 C
EEE 340
Lecture 04
12
2-5 Integrals Containing Vector Functions
.
Lin e in t egral   Vect o r

Surface in t egral 
 Scalar
Vo lum e in t egral


VF d v

CV dl
 
CF dl
 
S A ds
EEE 340
( 2  78)
( 2  79)
( 2  80)
( 2  81)
Lecture 04
13
The line integral around a close path C is denoted

as  Vd l
C
In the Cartesian coordinates (CRT)




 Vdl  C V ( x, y, z )ax dx  a y dy  az dz


 a x C V ( x, y, z )dx  a y C V ( x, y, z )dy

 az C V ( x, y, z )dz
EEE 340
Lecture 04
(2  82)
(2  83)
14
P

 r dr
• Example 2-13
2
O
• a) along the straight line OP, where P(1,1,0)
y
P(1,1,0)
P1
0
EEE 340
P2
Lecture 04
x
15
b). Along path OP1P
Solution. Using (2-52) of cylindrical
a). P 2   2 2
O r dr  ar O

 ar
r dr
 2
3
3
2 2 


(a x cos45  a y sin 45)
3
2  
 (a x  a y )
3
EEE 340
Lecture 04
16