fundamentals of electromagnetics
DESCRIPTION
co-ordinate transformationsTRANSCRIPT
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Fundamentals of Electromagnetics
Introductory ConceptsCoordinate Systems & Transformations
Vector Analysis
RICHU JOSE CYRIACNIT CALICUT
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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
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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
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Signals Amplification
Modulation
Antennas
“This part of the program is sponsored by” EM waves……….!!!!
Where Electromagnetic waves?
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•Maxwell’s equationEntire subject in one slide…!!!!!!
vD
0 B
t
BE
t
DJH
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Maxwell’s equation
Electrostatics(Only E-field)
Magnetostatics(Only H-field)
Electromagnetic waves
(both E&H field )
Fundamental laws of electromagnetics
0t
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» 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?????
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VECTOR ANALYSIS
Co-ordinate sytems CartesianCylindricalSpherical
Transformations of co-ordinate systemsVector Calculus
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Orthogonal and non-orthogonalNon orthogal is hard to work with….so we will discuss only
orthogonal co-ordinate systems
Co-ordinate systems
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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
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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
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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
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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
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zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ
CARTESIAN CYLINDRICAL SPHERICAL
ORDER Sradhikkanam…!!!
r,f, z r, q ,f
Summary
VECTOR REPRESENTATION: UNIT VECTORS
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Cartesian Coordinates
Differential quantities:Length:
Area:
Volume:
dzzdyydxxld ˆˆˆ
dxdyzsd
dxdzysd
dydzxsd
z
y
x
ˆ
ˆ
ˆ
dxdydzdv
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Cylindrical Coordinates
Differential quantities:
Length:
Area:
Volume:
dzzrddrrld ˆˆˆ
rdrdzsd
drdzsd
dzrdrsd
z
r
ˆ
ˆ
ˆ
dzrdrddv
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Distance = r sinq df
x
y
dfr sinq
r
f
P
x
z
y
q
Spherical Coordinates
PLEASE NOTE Bhaiyajiii….!!!.....its important
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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
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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
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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
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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
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TRANSFORMATIONS: SPHERICAL CYLINDRICAL
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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
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VECTOR ANALYSIS
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SCALAR FIELDS
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VECTOR FIELDS
Example: Velocity vector field - jet stream
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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
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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 )(
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‘ 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
)(
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T Known as Gradient of a scalar
A Known as Divergence of a vector