Medan Elektromagnetik
Electronics
Institut Teknologi Bandung
23 pag.SUWARNO SUWARNO
ELECTROMAGNETICS I ELECTROMAGNETICS I
EL 3001
Vector & Coordinate systems EL 3001
Vector & Coordinate systems
Vector and Scalar
• Scalar quantity : quantity that can be explained by only its magnitude
Ex: length, temperature, height, area, voltage, power
• Vector quantity : quantity can only
explained by its magnitude and direction.
Ex: velocity, acceleration, force, electric
field, magnetic field
Vektor Vektor
Vector quantity : expressed by its magnitude and direction.
a A
A A
A
a A
A A
A a A A
Unit Vector
VECTOR AND VECTOR COMPONENTS VECTOR AND VECTOR COMPONENTS
CARTESIAN
2 2
2
z y
x a a
a A
a A
z y
x
z y
x
A A A A
A A
A
2 2
A A x 2 A y A z
A
VEKTOR POSISI DAN VEKTOR JARAK VEKTOR POSISI DAN VEKTOR JARAK
-R 1
Position vector:
z 2 y
2 x
2 2
2
z 1 y
1 x
1 1
1
a a
a 0
R
a a
a 0
R
z y
x P
z y
x P
z 1 2
y 1 2
x 1 2
1 2
2 1 12
a ) (
a ) (
a ) (
R R
R
z z
y y
x x
P P
Distance Vector
2 1 2 1 / 2
2 1 2 2
1 2 12
) (
) (
) (
R
z z y
y x
x d
Distance
VECTOR ADDITION VECTOR ADDITION
C= A + B=B+A
VECTOR SUBSTRACTION VECTOR SUBSTRACTION
C= A-B= A +(- B)
B
C=A-B A
-B
VECTOR MULTIPLICATION VECTOR MULTIPLICATION
Multiplication by a scalar constant k
Same direction if k positive and opposite direction if k negative
A
kA
Longer if the absolute value of k >1, and shorter if
the absolute value of k <1
VECTOR PRODUCT VECTOR PRODUCT
dot product: the product of magnitude of the two vectors and cosine of the angle between the them.
A.B=B.A=AB cos AB
Projection A on B Length=A cos AB
2
B B . B A
cos :
Vektor
B
A AB B
DOT PRODUCT DOT PRODUCT
B A
B . cos A
vectors o
between tw Angle
A A
. A
) C(
. A B
. A C)
.(B A
) A(
. B B
. A
1 AB
2 2
A
ve distributi
e
commutativ
CROSS PRODUCT CROSS PRODUCT
CROSS PRODUCT CROSS PRODUCT
z y
x
z y
x
z y
x
B B
B
A A
A
a a
a AxA
AxB
a xa
a
; a
xa a
; a
xa a
0
ive) (distribut
C Ax AxB
C) Ax(B
ative) anticommut
( A Bx
AxB
y x
z x
z y
z y
x
TRIPPLE PRODUCT TRIPPLE PRODUCT
z y
x
z y
x
z y
x
C C
C
B B
B
A A
A C
C C
) x .(B A
meaning no
Ax(B.C)
l meaningful
A.(BxC)
!
! important is
Sequence
C.(AxB) A)
x .(
B )
x .(B A
C(A.B) -
B(A.C)
xC (AxB) Ax(BxC)
Determine
a. Vector A, its magnitude and unit vector of A
b. Angle that A makes with Y axis c. Angle between A and B
d. Perpendicular distance from the
origin to vector B
22
a 3 a
3 a
a 2 A ctor Unit ve
22 3
3 2
A a.Length
z y
x A
2 2
2
2 o
, 22 50
cos 3 A
a . cos A
axis Y
dan A
between b.Angle
y 1 1
-
1 o
, 27 145
22
3 15 cos 2
B A
B . cos A
B and A between c.Angle
1 1
-
68 , 2 )
180 (
sin OP
B to
O from Distance
d.
3 A o
COORDINATE SYSTEMS
• Cartesian (x,y,z)
• Cylindrical (z)
• Spherical (r,)
CARTESIAN
dy dl Z
dz
dx
X
Y
dS y =dx dz a y dS z =dx dy a z
dS x =dy dz a x
dv=dxdy dz
CYLINDRICAL CYLINDRICAL
CYLINDRICAL ELEMENTS CYLINDRICAL ELEMENTS
SPHERICAL SPHERICAL
SPHERICAL ELEMENT SPHERICAL ELEMENT
a d dR dS
a d dR sinθ
dS
a d dθ sinθ
dS R 2 R
R R R
Area elements
Base Unit Vector
obey right hand cyclic relation Base Unit Vector
obey right hand cyclic relation
a a
a a
a x a
a
a x a
a a
a a
a x a
a
a x a
a a
a a
a x a
a
a x a
r r
z z
z
y x
z x
z y
z y
x
r
x
x
x
