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co-ordinate transformations
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Fundamentals of Electromagnetics
Introductory ConceptsCoordinate Systems & Transformations
Vector Analysis
RICHU JOSE CYRIACNIT CALICUT
Why Study Electromagnetics? Electromagnetics is everywhere!!!It’s around you ..hmm…but you cannot detect
It’s the basic from which circuit theory is developed
Maxwell’s Equations Kirchoff’s laws
Its essential for a communication, signals are sent as Electromagnetic waves
What is the basic of electromagnetics? CHARGEElectromagnetics is the study of CHARGES
Electromagnetic study can be divided into three
o Electrostatics : charges are at restoMagnetostatics : charges are at steady motiono Electrodynamics : charges are in time varying motion(gives rise to wave that propagate and carry energy and information)
What is Electromagnetics?
rest motion
Signals Amplification
Modulation
Antennas
“This part of the program is sponsored by” EM waves……….!!!!
Where Electromagnetic waves?
•Maxwell’s equationEntire subject in one slide…!!!!!!
vD
0 B
t
BE
t
DJH
Maxwell’s equation
Electrostatics(Only E-field)
Magnetostatics(Only H-field)
Electromagnetic waves
(both E&H field )
Fundamental laws of electromagnetics
0t
» Is it difficult to study?
No…No…Never…..It’s interesting…..
NB: Maths is just a powerful tool…..Physical interpretation of the mathematical result is the key…
»We will acquire the tool for so called “OPERATION EMT” first….OK?
VECTOR ANALYSIS
How to study?????
VECTOR ANALYSIS
Co-ordinate sytems CartesianCylindricalSpherical
Transformations of co-ordinate systemsVector Calculus
Orthogonal and non-orthogonalNon orthogal is hard to work with….so we will discuss only
orthogonal co-ordinate systems
Co-ordinate systems
Orthogonal Coordinate Systems: (coordinates mutually perpendicular)
Spherical Coordinates
Cylindrical Coordinates
Cartesian Coordinates
P (x,y,z)
P (r, θ, Φ)
P (r, θ , z)
x
y
zP(x,y,z)
θ
z
rx y
z
P(r, θ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Selection of co-ordintes depends upon the symmetry of problem
Engineers always choose the easy way
x
z
y
yaxa
za
Unit Vector Representation for
Rectangular Coordinate System
xaThe Unit Vectors imply :
ya
za
Points in the direction of increasing x
Points in the direction of increasing y
Points in the direction of increasing z
Cartesian Coordinate System
VECTOR REPRESENTATION: UNIT VECTORS
zzyyxx aAaAaAA ˆˆˆ
r
f
z
P
x
z
y
Cylindrical Coordinate System
za
a
ra
The Unit Vectors imply :
za
Points in the direction of increasing r
Points in the direction of increasing j
Points in the direction of increasing z
ra
a
VECTOR REPRESENTATION: UNIT VECTORS
zzrr aAaAaAA ˆˆˆ
Spherical Coordinate System
r
f
P
x
z
y
q
a
a
ra
The Unit Vectors imply :Points in the direction of increasing r
Points in the direction of increasing j
Points in the direction of increasing q
ra
aa
VECTOR REPRESENTATION: UNIT VECTORS
aAaAaAA rr ˆˆˆ
zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ
CARTESIAN CYLINDRICAL SPHERICAL
ORDER Sradhikkanam…!!!
r,f, z r, q ,f
Summary
VECTOR REPRESENTATION: UNIT VECTORS
Cartesian Coordinates
Differential quantities:Length:
Area:
Volume:
dzzdyydxxld ˆˆˆ
dxdyzsd
dxdzysd
dydzxsd
z
y
x
ˆ
ˆ
ˆ
dxdydzdv
Cylindrical Coordinates
Differential quantities:
Length:
Area:
Volume:
dzzrddrrld ˆˆˆ
rdrdzsd
drdzsd
dzrdrsd
z
r
ˆ
ˆ
ˆ
dzrdrddv
Distance = r sinq df
x
y
dfr sinq
r
f
P
x
z
y
q
Spherical Coordinates
PLEASE NOTE Bhaiyajiii….!!!.....its important
Spherical Coordinates
Differential quantities:
Length:
Area:
Volume:
dRRddRR
dldldlRld R
sinˆˆˆ
ˆˆˆ
RdRddldlsd
dRdRdldlsd
ddRRdldlRsd
R
R
R
ˆˆ
sinˆˆ
sinˆˆ 2
ddRdRdv sin2
dRdl
Rddl
dRdlR
sin
x
y
zP(x,y,z)
θ
z
rx y
z
P(r, θ, z)
θ
Φ
r
z
yx
P(r, θ, Φ) Cartesian CoordinatesP(x,y,z)
Spherical CoordinatesP(r, θ, Φ)
Cylindrical CoordinatesP(r, θ, z)
TRANSFORMATIONS
TRANSFORMATIONS: CYLINDRICAL CARTESIAN
θr
P(r, θ, z)
xy
z
POINT TRANSFORMATION: If u are given a point in one co-ordinate system and to convert it to another
zzryrx sincos
zzx
yyxr 122 tan
zz
yx
yxr
ˆˆ
cosˆsinˆˆ
sinˆcosˆˆ
VECTOR TRANSFORMATION: If u are given a vector in one co-ordinate system and to convert it to another
zz
ry
rx
ˆˆ
cosˆsinˆˆ
sinˆcosˆˆ
TRANSFORMATIONS: SPHERICAL CYLINDRICAL
AREA INTEGRALS
• integration over 2 “delta” distances
dx
dy
Example:
x
y
2
6
3 7
AREA = 7
3
6
2
dxdy = 16
Note that: z = constant
VECTOR ANALYSIS
SCALAR FIELDS
VECTOR FIELDS
Example: Velocity vector field - jet stream
SOME TOOLS FROM OUR MATHS TOOL KIT
DEL operator- An operator that we are going to do operations on scalars and vectors
zk
yj
xi
Does not have any significance its own, but have significance when it OPERATES
DEL OPERATOR-PHYSICAL INTERPRETATION
‘ T ’ be a scalar and we are operating on T
RESULT OF OPERATION?????
T is a vector in the direction of the most rapid change of T,
and its magnitude is equal to this rate of change
If u substitute a point we get a direction in which a maximum variation from that point occurs
jxiyz
xyk
y
xyj
x
xyiT
)()()(xyxyzTLet )(
‘ T ’ be a vector and we are operating on T
RESULT OF OPERATION????? T is the net flux of T per unit volume at the point considered, countingvectors into the volume as negative, and vectors out of the volume as positive.
DEL OPERATOR-PHYSICAL INTERPRETATION
z
T
y
T
x
T
kTjTiTz
ky
jx
iT
zyx
zyx
)(
T Known as Gradient of a scalar
A Known as Divergence of a vector