Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
COORDINATE SYSTEMS & GAUSS'S LAW
by
Dr. Sikder Sunbeam Islam
Dept. of EEE.
IIUC
Course Title: Engineering Electromagnetism
SYLLABUS (UP TO MID-TERM) : 30 MARKS
2
WHAT IS ELECTROMAGNETISM?
Electromagnetism is a branch of physics which deals
with electricity and magnetism and the interaction
between them.
Electromagnetism is basically the science of
electromagnetic fields. An electromagnetic field is the
field produced by objects that are charged electrically.
Radio waves, infrared waves, Ultraviolet waves, and
x-rays are all electromagnetic fields in a certain range
of frequency.
3
VECTOR BASICS
Vector: A quantity with both magnitude and
direction. Example: Force.
Scalar: A quantity that does not posses direction .
Example: Temperature.
4
VECTOR BASICS :CONTINUES….
Vector Addition:
Vector Subtraction:
Vector Multiplication:
5
Unit vectors: , and directed along x, y,
and z respectively with unity length and no
dimensions.
The vector r=A+B+C may be written in
terms of unit vectors as:
r =A+B+C = A +
Where:
A is the directed length or signed
magnitude of A.
Example: A unit vector in the direction of B is;
VECTOR BASICS :CONTINUES….
6
Dot Product
Which results in a scalar value, and is the smaller
angle between A and B.
VECTOR BASICS :CONTINUES….
7
Vector Basics :continues….
8
VECTOR BASICS :CONTINUES….
Dot products of two vectors A and B is, 9
VECTOR BASICS :CONTINUES….
10
VECTOR BASICS :CONTINUES….
11
Problem: a)Write the expression of the vector going from point
P1(1,3,2) to point P2(3,-2,4) in Cartesian coordinate. b) What is the
length of this line?
VECTOR BASICS :CONTINUES….
12
VECTOR FIELD
A vector field in the plane, can be visualized as a
collection of arrows (flux lines) with a given magnitude
and direction, each attached to a point in the plane.
A vector filed strength is measured by the number of flux
lines passing through a unit surface normal to the
vector.
Flux flow of vector is like flow of water or fluid.
A enclosed surface of a volume will have
outward/inward flow of flux through this surface when
the volume contains source/sink.
The net outward flow per unit volume is therefore the
measure of strength of the enclosed source.
In the uniform field, there is an equal amount of inward
and outward flux going through any closed volume
containing no source or sink , results zero divergence.
13
VECTOR FIELD
A kind of source called vortex source that causes
circulation of vector filed around it (source).
If A is a force acting on an object, its circulation
will be the work done by the force in moving the
object once around the contour.
Similarly, the phenomenon of water whirling down
a sink drain is an example of vortex sink.
14
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
15
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
A point is located by its and coordinates,
or as the intersection of three constant
surfaces (planes in this case) y , x, z 16
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
Increasing each coordinate
variable by a differential
amount dx, dy , and dz, one
obtains a parallelepiped. dx
dy dz
Differential volume: dv = dx dy dz
Differential Surfaces: Six planes with
dierential areas ds=dxdy; ds=dzdy; ds= dxdz
Differential length: from P to P’ ,
17
Line Integral [Application of scalar(dot)product ] Suppose, we move along a path from P1 to P2 in a radial force field F. F acting in the r
direction. At any point P, the product of path length dL (incremental) and
component of F parallel to it is given by,
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
Component of dL in the r direction is, dr=cosϴ dL i.e.[cosϴ=dr/dL]
i.e.[cosϴ=FL/F]
Using vector (dot product),
If work dW done by force F moving an object a distance dr= cosϴ dL then,
dW=F.dL= Fcosϴ dL . Total work W,
[This formulation is called line integral.
[Work for path.]
18
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
PROBLEM: Find the work required to move a 5KG mass from x=0,
y=0 to x=8, y=7 against a force . Ans=171J.
19
Surface Integral:
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
We can express the flow of water, Ψ=BA cosθ=B.A (for Uniform)
Integrating the contribution of all
points across the surface of the loop,
obtaining the total flow of water ,
(for nonuniform)
20
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
21
Volume Integral: Example (Solve yourself)
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
Ans. 1440 kg
22
23
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
The Divergence of A at a given point P is the outflow of flux
from a small closed surface per unit volume as the volume
shrinks to zero.
24
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
The curl of A is an axial(rotational) vector whose magnitude is the
maximum circulation of A per unit area as the area tends to zero and
whose direction is the normal direction of the area.
25
COORDINATE SYSTEM :
RECTANGULAR (CARTESIAN) COORDINATE
The divergence theorem states that the total outward flux of a vector
field A through the closed surface S is the same as the volume
integral of the divergence of A.
26
The Stoke’s theorem proposing that the surface integral of the curl
of a vector field A over any surface bounded by a closed path is equal
to the line integral of a vector field A round that path.
COORDINATE SYSTEM :
CYLINDRICAL COORDINATE
[In some books is
expressed with r ]
27
COORDINATE SYSTEM :
CYLINDRICAL COORDINATE
28
COORDINATE SYSTEM :
SPHERICAL COORDINATE
29
COORDINATE SYSTEM :
SPHERICAL COORDINATE
30
COORDINATE SYSTEMS
31
COORDINATE SYSTEM : TRANSFORMATIONS BETWEEN
CYLINDRICAL AND CARTESIAN COORDINATES
32
COORDINATE SYSTEM : TRANSFORMATIONS BETWEEN
CYLINDRICAL AND CARTESIAN COORDINATES
OR,
33
COORDINATE SYSTEM : TRANSFORMATIONS BETWEEN
CYLINDRICAL AND CARTESIAN COORDINATES
Example-2: Transform the vector
in Cartesian Coordinates.
34
COORDINATE SYSTEM : TRANSFORMATIONS BETWEEN
SPHERICAL AND CARTESIAN COORDINATES
35
COORDINATE SYSTEM : TRANSFORMATIONS BETWEEN
SPHERICAL AND CARTESIAN COORDINATES
36
COORDINATE SYSTEM : TRANSFORMATIONS BETWEEN
SPHERICAL AND CARTESIAN COORDINATES
37
Electrostatic Field
Topics to be covered
(Ref. Chapter-4)
• Article 4.1-4.4 (Basic concept of Electric field)
• Article 4.5-Gauss’s law specifically Maxwell’s equation
• Article 4.6-Application of Gauss’s Law
• Article 4.7-Electric Potential
• Article 4.8- Relationship between E & V
• Article 4.9-An Electric Dipole & Flux Lines
• Article 4.10-Energy Density in Electrostatic Fields
38
Basic concept of Electric field
Electric charge (Q) is the physical properties of matter
that cause it to experience a force when placed in an
electromagnetic field.
Point Charge means a charge that is located on a body
whose dimensions are much smaller than the relevant
dimension.
Coulomb’s Law: This law states that, two point charge
Q1 and Q2 separated by a distance R experience a force,
So,
39
Basic concept of Electric field: cont….
Electric Field Intensity (E) is the force per
unit charge when placed in the electric field.
So,
Electric Fields due to continuous charge distribution
40
Electric Flux Density (D)
Basic concept of Electric field: cont….
So,
41
GAUSS’S LAW-MAXWELL’S EQUATION
Gauss’s law states that, the total electric flux through
any closed surface is equal to the total charge enclosed by the
surface.
-------------(1)
Charge density
42
Now applying Divergence theorem ( ) to equ.(1),
GAUSS’S LAW-MAXWELL’S EQUATION
-------------(2)
Now comparing eqt.(1) and (2) we find,
-------------(3)
Which is the first of four Maxwell’s Equations to be derived. It states
that a volume charge density is the same as Divergence of Electric
flux density.. Is the charge per unit volume.
1. Equation 1 & 3 states the same Gauss’s law in different way (equ.1 is
integral form whereas equ.3 is differential form)
2. Gauss's law is an alternative statement of Coulomb's law; proper application
of the divergence theorem to Coulomb's law results in Gauss's law.
3. Gauss's law provide* an easy means of finding E or D for symmetrical
charge distributions such as a point charge, an infinite line charge, an infinite
cylindrical surface charge, and a spherical distribution of charge.
Points to be noted
43
Gauss’s law-Maxwell’s equation
44
APPLICATIONS OF GAUSS'S LAW
Gauss's law to calculate the electric field
involves first knowing whether symmetry
exists.
If field is symmetric, we construct a
mathematical closed surface (known as a
Gaussian surface).
The surface is chosen such that D is normal or
tangential to the Gaussian surface.
if D is tangential, then D. dS=D.ds, as D is
constant on the surface.
if D is normal, then D. dS=0.
45
APPLICATIONS OF GAUSS'S LAW: POINT CHARGE
Suppose, a point charge Q located at the origin of spherical
coordinate system. To determine D at a point P it is easy to assume a
closed surface of radius r=a containing P.
-------------(1) At, r=a
P
46
APPLICATIONS OF GAUSS'S LAW: UNIFORMLY
CHARGED SPHERE
-------------(1)
47
APPLICATIONS OF GAUSS'S LAW: UNIFORMLY
CHARGED SPHERE-CONT….
or,
while,
-------------(2)
-------------(3)
-------------(4)
Now from equ.4 and 5,
-------------(5) 48
APPLICATIONS OF GAUSS'S LAW: UNIFORMLY
CHARGED SPHERE-CONT….
-------------(6)
Now from equ.3 and 6,
49
APPLICATIONS OF GAUSS'S LAW: UNIFORMLY
CHARGED SPHERE-CONT….
50
APPLICATIONS OF GAUSS'S LAW
Example:
or,
Apply Gauss Law,
51
APPLICATIONS OF GAUSS'S LAW
Example:
52
ELECTRIC POTENTIAL
-------------(1)
-------------(2)
Dividing W by Q gives potential energy per unit charge
Denoted by as potential difference between A and B points.
-------------(3)
If E field in Fig. is due to point charge Q located at origin, then -------(4)
Now from, equ.3
Then, Potential,
53
ELECTRIC POTENTIAL: EXAMPLE
Hence,
54
RELATIONSHIP BETWEEN E AND V-MAXWELL’S
EQUATIONS
Now we know, the potential difference between points A and B is independent
of the path taken. Hence,
-------------(1)
This shows that, the line integral of E along a closed path must be zero. Physically it
reveals that no net work is done in moving a charge along a closed path in electrostatic
field. Applying Stokes’s Theorem in equ.(1) ,
-------------(2)
55
Vectors whose line integral does not depends on the path of integral are called
conservative vectors. Thus Electrostatic field is a conservative field. Equ.
(1) or (2) referred to as Maxwell’s Equation. Equ(1) is integral form and
equ(2) id differential form.
RELATIONSHIP BETWEEN E AND V-MAXWELL’S
EQUATIONS
56
Example:
57
58
AN ELECTRIC DIPOLE & FLUX LINES
When two point charges of equal magnitude but opposite sign are
separated by a small distance then an Electric Dipole is formed.
When r1 and r2 is the distance
between P and +Q and –Q
respectively. If dipole moment
is p then, the potential at
origin is,
An electric flux line is an imaginary path or line drawn in
such a way that its direction at any point is the direction of
the electric field at the point.
Note that the dipole moment p is directed from -Q
to +Q. If the dipole center is not at the origin but at
r', then equtn. Becomes,
59
EXAMPLE
60
ENERGY DENSITY IN ELECTROSTATIC FIELDS
Suppose we wish to position three point charges Q1, Q2, and Q3
in an initially empty space shown shaded in Figure.
Hence the total work done in positioning the three charges is,
where V1, V2, and V3 are total potentials at P1, P2, and P3, respectively.
In general, if there are n point charges, eq. (1) becomes
Fig. Assembling of
charges
61
ENERGY DENSITY IN ELECTROSTATIC FIELDS
According to Eqn.(2),
So, from Eqn.(4),
By applying divergence theorem to the first term at right side
while dS varies as . Consequently, the first integral in eq. (6) must tend to zero as
the surface S becomes large. Hence, eq. (6) reduces to
62
EXAMPLE
63
Alternatively
REFERENCE
Engineering Electromagnetics; William Hayt &
John Buck, 7th & 8th editions; 2012
Electromagnetics with Applications, Kraus and
Fleisch, 5th edition, 2010
Elements of Electromagnetics ; Matthew N.O.
Sadiku
64