VECTOR ANALYSIS
REVIEW
• Dot Product
A . B = [A][B] cosθAB
• Cross Product
A X B = aN [A][B] sinθAB
PROBLEM 1
• Given the two vectors , rA = - ax – 3ay – 4az and rB =
2ax + 2ay + 2az, and point C(1,3,4).
• Find:
(a)
RAB; (b) [rA]; (c) aA; (d) aAB;
(e) a unit vector directed from C toward A.
(a)
(b)
(c)
(d)
(e)
pp. 9
3ax + 5ay +6az
5.10
– 0.196ax – 0.588ay – 0.784az
0.359ax + 0.598ay +0.717az
– 0.196ax – 0.588ay – 0.784az
PROBLEM 2
The three vertices of a triangle are
located at A(6, -1, 2), B(-2, 3, -4) and
C(-3, 1, 5).
Find :
(a) RAB X RAC
(b) The area of the triangle
(a) 24ax + 78ay +20az
(b) 42.0
PROBLEM 3
Given points A(2, 5, -1), B(3, -2, 4) and C(-2, 3, 1).
Find :
(a) RAB . RAC
(b) The angle between RAB and RAC
(c) The length of the projection of RAB on RAC
(d) The vector projection of RAB on RAC
(a)
(b)
(c)
(d)
pp.12
20
61.9 degrees
4.08
-3.33ax – 1.667ay + 1.667az
PROBLEM 4
Given points A(8,−5, 4) and B(−2, 3, 2),
find:
a)
the distance from A to B.
|B − A| = |(−10, 8,−2)| = 12.96
b)
A unit vector directed from A towards B.
c)
a unit vector directed from the origin to the
midpoint of the line AB.
1.4
COORDINATE SYSTEMS
COORDINATE SYSTEMS
x = ρcosǾ
Y = ρsinǾ
Z=z
COORDINATE SYSTEMS
x = rsinθcosǾ
y = rsinθ sinǾ
z = rcosθ
ELECTRIC CHARGE
AND
COULOMB’S LAW
INTRODUCTION TO ELECTROMAGNETIC
FIELDS
• Electromagnetics is the study of the effect of
charges at rest and charges in motion.
• Some special cases of electromagnetics:
• Electrostatics: charges at rest
• Magnetostatics: charges in steady motion (DC)
• Electromagnetic waves: waves excited by
charges in time-varying motion
14
INTRODUCTION TO
ELECTROMAGNETIC FIELDS
• transmitter and receiver
are connected by a “field.”
15
INTRODUCTION TO
ELECTROMAGNETIC FIELDS
• When an event in one place has an effect on something
at a different location, we talk about the events as being
connected by a “field”.
• A field is a spatial distribution of a quantity; in general, it
can be either scalar or vector in nature.
16
INTRODUCTION TO
ELECTROMAGNETIC FIELDS
• Electric and magnetic fields:
•
•
•
17
Are vector fields with three spatial components.
Vary as a function of position in 3D space as well as time.
Are governed by partial differential equations derived from
Maxwell’s equations.
FUNDAMENTAL CHARGE: THE
CHARGE ON ONE ELECTRON.
e = 1.6 x 10 -19 C
Unit of charge is a Coulomb (C)
TWO TYPES OF CHARGE:
Positive Charge: A shortage of electrons.
Negative Charge: An excess of electrons.
Conservation of charge – The net charge of a
closed system remains constant.
ELECTRIC FORCES
Like Charges - Repel
F
+
+
Unlike Charges - Attract
-
F
F
+
F
ELECTROMAGNETICS
IN
3D PLANE
INTRODUCTION TO ELECTROMAGNETIC
FIELDS
• Universal constants in electromagnetics:
• Velocity of an electromagnetic wave (e.g.,
light) in free space (perfect vacuum)
c  3  10 m/s
8
• Permeability of free space
 0  4  10 H/m
7
• Permittivity of free space:
 0  8.854  10
12
F/m
• Intrinsic impedance of free space:
 0  120 
23
INTRODUCTION TO ELECTROMAGNETIC
FIELDS
• Relationships involving the universal constants:
c
1
 0 0
In free space:
B  0 H
24
D  0 E
0
0 
0
Example
Two charges are separated by a distance r and have a force
F on each other.
q1q2
F k 2
r
F
q2
q1
F
r
If r is doubled then F is :
¼ of F
If q1 is doubled then F is :
2F
If q1 and q2 are doubled and r is halved then F is : 16F
Example
Three charged objects are placed as shown. Find the net
force on the object with the charge of -4μC.
F  k
- 5μC
45º
20cm
202  202  28cm
q1q 2
r2
(5 106 )(4 106 )
F1  9 10
 4.5N
2
(0.20)
9
(5106 )(4106 )
F2  910
 2.30N
2
(0.28)
9
5μC
F1 45º
- 4μC
20cm
F2
F1 and F2 must be added together as vectors.
COULOMB’S LAW
• A colonel in the French Army Engineers, Col. Charles Coulomb,
stated that the force between two very small objects separated in
a vacuum or free space by a distance which is large compared to
their size is proportional to the charge on each and inversely
proportional to the square of the distance between them.
q1q2
F k 2
R
𝑘
1
4𝜋Ԑo
Where:
Q1, Q2 are the charges
R is the separation
K is the proportionality constant
Ԑo is the permittivity of free space
∈𝑜
8.854𝑥10
1
𝐹
10 ,
36𝜋
𝑚
RECALL VECTOR ANALYSIS
• 3D plane of a cartesian
coordinate system
If there exist a point P and
Q, respectively with
corresponding coordinates
as seen in the figure.
RECALL VECTOR ANALYSIS
• 3D plane of a cartesian
coordinate system
At point P:
rp  ax  2 ay  3az
At point Q:
rQ  2 ax - 2 ay  az
Resultant vector is ͞RPQ
RPQ  rQ - rP
RPQ  (2 -1)ax  (-2 - 2)ay  (1- 3)az
RPQ  ax - 4ay - 2az
RECALL VECTOR ANALYSIS
• 3D plane of a cartesian
coordinate system
At Resultant vector is ͞RPQ
RPQ  ax - 4ay - 2az
The scalar distance RPQ
RPQ  12   4    2 
2
RPQ  21
2
THE EXPERIMENTAL LAW OF COULOMB
• 3D plane of a cartesian
coordinate system
If there exist a charge on both points Q
and P being -2μC and 5μC respectively.
Then the force between the two
charges is:
Q QQ P
a PQ
2
r
Q QQ P

a PQ
2
4  o | R PQ |
F PQ  k
F PQ
Where:
 o  8.854 X 1012
C
Nm 2
2
aPQ  a unit vector in the direction of RPQ
RPQ
ax  4ay  2az


| RPQ |
21
The Experimental Law of Coulomb
Solving for the force experienced by the charge on point Q:
Q QQ P
FQ 
a PQ
2
4  o | R PQ |
(  2 x10  6 C )(5 x10  6 C )  ax  4 ay  2 az 
FQ 

 12
2 
4 (8.854 x10 )( 21) 
21

 ax  4 ay  2 az 
F Q  4.28 x10 3 
, N
21


 ax  4 ay  2 az 
F P  4.28 x10 
, N
21


3
The force expressed by
Coulomb’s law is a mutual
force, for each of the two
charges experiences a force
of the same magnitude,
although of opposite
direction.
PROBLEM
Given a charge of 3 x 1 0  4 C at P(1, 2, 3) and a charge of
 1 X 10 4 C at Q(2, 0, 5) in a vacuum.
Determine the Force on charge Q due to P.
ax  2ay  2az
FQ  30(
), N
3
FP   FQ
FP  10ax  20ay  20az , N
pp.29
PROBLEM
A 2-mC positive charge is located in a vacuum at P1(3, -2, -4), and a
5-μC negative charge is at P2(1, -4, 2).
Find:
a. The vector force on the negative charge
b. The magnitude of the force on the charge at P1
F2  0.616ax  0.616a y  1.848az , N
F1  2.04, N
pp.30
PROBLEM
Point charges of 50nC each are located at A(1, 0, 0), B(−1, 0, 0),
C(0, 1, 0), and D(0,−1, 0) in free space.
Find the total force on the charge at A.
where RCA = ax −ay , RDA = ax +ay , and RBA = 2ax . The
magnitudes are |RCA| = |RDA| = 2 , and | RBA| = 2.
𝐹
50𝑥10
4𝜋𝜀𝑜
𝐹
50𝑥10
4𝜋𝜀𝑜
𝐹
21.5𝑎𝑥𝜇𝑁
2.3
𝑅𝐷𝐴
|𝑅𝐷𝐴|
𝑅𝐶𝐴
|𝑅𝐶𝐴|
1
1
2 2
2 2
2
𝑎𝑥
8
𝑅𝐵𝐴
|𝑅𝐵𝐴|
ELECTRIC FIELD INTENSITY
Recall Coulomb’s Law Problem: If a 2-mC positive charge is located in a
vacuum at P1(3, -2, -4), and a 5-μC negative charge is at P2(1, -4, 2).
If the charge on P1 is made stationary, and moving the charge on P2 slowly
around the charge on point P1, there exist everywhere a force on the second
charge; The second charge is now displaying the existence of a force field.
E
(5 x10  6 C )
4 (8.854 x10
 12
)

21

2
 ax  4 ay  2 az  V

; m
21


 ax  4 ay  2 az  V
E  2139.94 
, m
21


ELECTRIC FIELD INTENSITY
“the vector force on a unit positive test charge”
Q SQ t
F St 
a St
2
4 o | R St |
F St
QS
a St

2
Q t 4 o | R St |
J Nm
Q
N V C C
E
a
R
; ; ; ;
4 o | R |2
C m m m
ELECTRIC FIELD INTENSITY
Let a point charge Q1, 25nC be at P1(4, -2, 7) and a charge Q2, 60nC be at P2
(-3, 4, -2). Find E at P3 (1, 2, 3), if ε = εo.
ELECTRIC FIELD INTENSITY
If there exist a multiple 3nC point charges as shown in the figure, solving for the
total electric field experienced by point P.
FIELD OF A LINE CHARGE
L
E
a
2 o 
w here :
 L  is the line charge density

 x  6    y  8
2
2
SAMPLE PROBLEMS
1)
Infinite uniform line charges of 5nC/m lie along the (positive and negative) x
and y axes in free space. Find E at: a) (0, 0, 4), b) (0, 3, 4)
2)
An infinitely long uniform line charge is located at Y = 3, Z = 5. If 𝜌𝐿 =
30nC/m, find E at: a) the origin; b) PB(0, 6, 1); c) PC(5, 6, 1).
3)
A uniform line charge of 2µC/m is located on the z – axis. Find E in
Cartesian coordinates at P(1, 2, 3) if the charge extends from - ∞ < z < ∞.
4)
Uniform line charges of 120 nC/m lie along the entire extent of the three
coordinate axes. Assuming free space conditions, find E at P(-3, 2, -1).
FIELD OF A SHEET OF CHARGE
w here :
 S  is the field of a sheet of charge
SAMPLE PROBLEMS
1)
Three infinite uniform sheets of charge are located in free space as
follows: 3 nC/𝑚 at z = -4, 6 nC/𝑚 at z = 1, and -8 nC/𝑚 at z = 4.
Find E at the points: (a) PA(2,5,-5); (b) PB(4,2,-3); (c) PC(-1,-5,2); (d)
PD(-2,4,5).
2)
Two identical uniform sheet charges with ρs = 100 nC/𝑚 are located
in free space at z = 2cm. What force per unit area does each sheet
exert on the other?
3)
Given the surface charge density, 𝜌s = 2 𝜇C/m , existing in the region
𝜌 0.2m, z = 0, find E at a) PA(𝜌 = 0, z = 0.5); b) PB (𝜌 = 0, z = -0.5).
FIELD DUE TO A CONTINUOUS VOLUME
CHARGE DISTRIBUTION
The small amount of charge ΔQ in a small volume Δv is
ΔQ = ρv Δv
The total charge within some finite volume is obtained by
integrating throughout that volume
Q    vdv
vol
Q   v   d  d dz
Q   v  r 2 sin  drd d
SAMPLE PROBLEMS
1)
A cylindrical volume contains a uniform volume charge density of 1x10
C/m . Calculate the charge within the region 0 ≤ 𝜌 ≤ 4, -𝜋/2 ∅ 𝜋/2, 10 z 10;
2)
A uniform volume charge density of 0.2μC/m3 is present throughout the
spherical shell extending from r = 3 cm to r = 5 cm. If ρν = 0 elsewhere, find
(a) the total charge present throughout the shell, and (b) r1 if half the total
charge is located in the region 3 cm < r < r1.
3)
A spherical volume having a 2-μm radius contains a uniform volume charge
density of 1X10 C/m3. (a) What total charge is enclosed in the spherical
volume? (b) Now assume that a large region contains one of these little
spheres at every corner of a cubical grid 3 mm on a side and that there is no
charge between the spheres. What is the average volume charge density
throughout this large region?
ELECTRIC FIELD INTENSITY
A point charge Q1 = 25nC at (4, -2, 7), a uniform line charges of
120 nC/m lie along the entire extent of the three coordinate axes
and a volume charge density at origin, ρv = 10 C/m enclosed
in the region 4μ m < r < 5μm, 0 < θ < 25 and 0.9𝜋 < 𝜙 <
1.1𝜋. A sheet charge of 20 nC/m at x = 3, Assuming free space
conditions, find E at P( 5, 63.435 , 3).
1)
Uniform line charges of 120 nC/m lie along the entire
extent of the three coordinate axes. Assuming free
space conditions, find E at P(-3, 2, -1).
2)
A uniform line charge density of 180 nC/m lies along
the x – axis. A sheet of charge equal to 20nC/sq. m
lies in the z = 1 plane, a point charge of 6μC is
located at the origin.
Find the total Field at P(2,3,4), in cartesian, cylindrical
and spherical coordinates.
A charge Qo, located at the origin in free space, produces a field for
which Ez = 1kV/m at point P(-2, 1, -1).
a)
Find Qo
b)
E at M(1, 6, 5) in Cartesian, cylindrical and spherical
coordinates.