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CHAPTER 1:
Vector Fields
ELECTROMAGNETIC FIELDS
THEORY
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Scalar
Scalar:A quantity that has only magnitude.
For example time, mass, distance,
temperature and population are scalars. Scalar is represented by a lettere.g., A, B
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Vector
Vector: A quantity that has both magnitudeand
direction.
Example: Velocity, force, displacement and electric
field intensity. Vector is represent by a letter such asA, B, or
It can also be written as
where A is which is the magnitude and is unitvector
aAA
A
a
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Unit Vector
A unit vector along A is defined as a vector whosemagnitude is unity (i.e., 1) and its direction is alongA.
It can be written as aAor
Thus
a
A
A
aAA
A
A
A
Aa
||
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Vector Subtraction
Vector subtraction is similarly carried out as
D=AB=A+ (-B)
A
B
BA
B
Figure (a)
Figure (b)
Figure (c)
Figure (c) shows that vector D is a vector
that is must be added to Bto give vectorASo if vectorA and Bare placed tail to tail
then vector Dis a vector that runs from the
tip of BtoA.
A
B
BA
ABA
B
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Vector multiplication
Scalar (dot ) product (AB)
Vector (cross) product (A X B)
Scalar triple productA (B X C)
Vector triple productA X (B X C)
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Multiplication of a vector by a scalar
Multiplication of a scalar kto a vectorAgives a vectorthat points in the same direction asAand magnitude equal
to |kA|
The division of a vector by a scalar quantity is a
multiplication of the vector by the reciprocal of the scalar
quantity.
AkA
Ak1|| k
1|| k
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Scalar Product
The dot product of two vectors and , written asis defined as the product of the magnitude of and , and
the projection of onto (or vice versa).
Thus ;
Where is the angle between and . The result of dotproduct is a scalar quantity.
BA
cos|||| BABA
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Vector Product
The cross (or vector) product of two vectors A and B,written as is defined as
where; a unit vector perpendicular to the plane thatcontains the two vectors. The direction of is taken asthe direction of the right thumb (using right-hand rule)
The product of cross product is avector
nBABA AB sin||||
nn
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Right-hand Rule
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Dot product
If and then
which is obtained by multiplying A and B
component by component.
It follows that modulus of a vector is
zzyyxx BABABABA
),,( zyx BBBB
),,( zyx AAAA
222|| zyx AAAAAA
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Cross Product IfA=(Ax, Ay, Az), B=(Bx, By, Bz) then
zyx
yx
yxz
xz
xzy
zy
zyx
zyx
zyx
aBB
AAa
BB
AAa
BB
AA
BBB
AAA
aaa
BA
zxyyxyzxxzxyzzy aBABAaBABAaBABA )()()(
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Cross Product
Cross product of the unit vectors yield:
yaz
xzy
zyx
aaa
aaa
aaa
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Example 1
Given three vectors P=
Q=
R=
Determine
a) (P+Q) X (P-Q)
b) Q(R X P)
c) P(Q X R)
d) e) P X( Q X R)
f) A unit vector perpendicular to both Q and R
zx
aa 2zyx aaa 22
zyx aaa 32
QRsin
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Solution (cont)
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To find the determinant of a 3 X 3 matrix, we repeat the first two rows andcross multiply; when the cross multiplication is from right to left, the result
should be negated as shown below. This technique of finding a determinant
applies only to a 3 X 3 matrix. Hence
Solution (cont)
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Solution (cont)
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Solution (cont)
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Solution (cont)
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Cylindrical Coordinates
Very convenient when dealing with problems havingcylindrical symmetry.
A point P in cylindrical coordinates is represented as (,, z) where
: is the radius of the cylinder; radial displacement from the z-axis
: azimuthalangle or the angular displacement from x-axis
z: vertical displacement z from the origin (as in thecartesian system).
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a
a
za
Cylindrical Coordinates
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Cylindrical Coordinates
The range of the variables are
0 < , 0 < 2, - < z <
vector in cylindrical coordinates can be writtenas (A,A, Az) or Aa+ Aa+ Azaz
The magnitude of is
222|| zAAAA
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Relationships Between Variables
The relationships between the variables (x,y,z) of the
Cartesian coordinate system and the cylindrical system (,
, z) are obtained as
So a point P (3, 4, 5) in Cartesian coordinate is the same
as?
zz
xy
yx
/tan 1
22
zz
y
x
sin
cos
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Relationships Between Variables
So a point P (3, 4, 5) in Cartesian coordinate is the same asP ( 5, 0.927,5) in cylindrical coordinate)
5
927.03/4tan
543
1
22
z
rad
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Spherical Coordinates (r,,)
The spherical coordinate system is useddealing with problems having a degree ofspherical symmetry.
Point P represented as (r,,) where r : the distance from the origin,
: called the colatitude is the angle between z-axis andvector of P,
: azimuthal angle or the angular displacement fromx-axis (the same azimuthal angle in cylindricalcoordinates).
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Spherical Coordinates
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The range of the variables are
0 r< , 0 < , 0 < < 2
A vectorAin spherical coordinates written as(Ar,A,A) or Arar+ Aa+ Aa
The magnitude of A is
222|| AAAA r
Spherical Coordinates (r,,)
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Relation to Cartesian coordinates system
x
y
zyx
zyxr
1
221
222
tan
)(tan
cos
sinsin
cossin
rz
ry
rx
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Relationship between cylinder and spherical
coordinate system
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Point transformation between cylinder and spherical
coordinate is given by
or
Point transformation
22 zr z 1tan
sinr cosrz
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Express vector B=
in Cartesian and cylindrical coordinates. Find Bat (-3,
4 0) and at (5, /2, -2)
Example
aaa cos10 rr
r
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Differential Elements
In vector calculus the differential elementsin length,
area and volumeare useful.
They are defined in the Cartesian, cylindrical and
spherical coordinate
Di
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Cartesian Coordinates
P Q
S R
dy
dz
dx
A B
D C
xx
y yax ay
az
z
Differential displacement : zyx dzadyadxald
fferentialelements
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z
y
x
dxdySd
dxdzSd
dydzSd
a
a
a
Differential normal area:
Cartesian Coordinates
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Differentialdisplacement
Differential
normal area
Differentialvolume
zyx dzdydxld aaa
z
y
x
dxdySd
dxdzSd
dydzSd
a
a
a
dxdydzdv
Cartesian Coordinates
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Cylindrical Coordinates
Differential displacement :zdzddld aaa
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Differential normal area:
zaddSd
dzadSd
dzadSd
Cylindrical Coordinates
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Differentialdisplacement
Differential normal
area
Differential volume
zdzaadadld
zaddSd
dzadSd
dzadSd
dzdddv
Cylindrical Coordinates
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Spherical Coordinates
drd sin
Differential displacement : adrarddrald r sin
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a
a
a
rdrdSd
drdrSd
ddrSdr
sin
sin2
Differential normal area:
Spherical Coordinates
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Differentialdisplacement
Differential
normal area
Differentialvolume
adrarddrald r sin
ardrdSd
adrdrSd
addrSdr
sin
sin2
ddrdrdv sin2
Spherical Coordinates
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1. The gradient of a scalar V, written as V
2. The divergence of a vectorA, written as
3. The curl of a vectorA, written as
4. The Laplacian of a scalar V, written as
Written as is the vector differential operator. Alsoknown as the gradient operator. The operator in useful indefining:
Del Operator
A
A
V2
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Example :
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Divergence
In Cartesian coordinates,
In cylindrical coordinates,
In spherical coordinate,
z
z
y
y
x
x AAA
A
z
AAA
z
1)(
1A
A
rA
rAr
rrA r
sin
1)sin(
sin
1)(
1 22
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Example
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In Cartesian coordinates,
In cylindrical coordinates,
zyx
zyx
AAAzyx
A
aaa
Curl of a Vector
z
z
AAA
z
aaa
1A
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Curl of a Vector
In spherical coordinates,
ArrAA
r
arraa
r
r
r
)sin(
)sin(
sin
12
A
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Examples on Curl Calculation
)(cossin 2
xzxye
zyx
aaa
xy
zyx
A
y
A
x
A
x
A
z
A
z
A
y
A xyzxyzzyx aaaxA
xyxexyyxzxzz cossincos2000 zyx aaaxA
xyxexyyxzz cos2sin zy aaxA
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Laplacian of a scalar
The Laplacian of a scalar field V, written as 2V is defined
as the divergence of the gradient of V.
In Cartesian coordinates,
2
2
2
2
2
22
z
V
y
V
x
VV
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In cylindrical coordinates,
In spherical coordinates,
2
2
222
2
2
2 V
sinr
1Vsin
sinr
1
r
Vr
rr
1V
Laplacian of a scalar
2
2
2
2
2
2 11
z
VVVV
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References:
Richard S . Muller and Theodore I. KaminsDeviceElectronics for Integrated Circuits
Jacob Milman, Christos C. Halkias
IntegratedElectronics
Specialization is an art of knowing more and
more about less and less and finally knowingeverything about nothing - Zentill [ ]