Electromagnetic Waves and Transmission Lines UNIT-3-PPT.pdf
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About This Presentation
EMWTL
Slide Content
ADITYA ENGINEERING COLLEGE (A)
Electro Magnetic Waves And Transmission Lines
(UNIT-III)
Mr. T. SRINIVASARAO
Assistant Professor Department of Electronics and Communication
Engineering
Aditya Engineering College (A)
Email: [email protected]
Electro Magnetic Waves And Transmission Lines
(UNIT-III)
Mr. T. SRINIVASARAO
Assistant Professor Department of Electronics and Communication
Engineering
Aditya Engineering College (A)
Email: [email protected]
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr. T. SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
UNIT -III: ELECTROSTATICS & MAGNETOSTATICS (Syllabus )
Review on Co-ordinate Systems, Vector Calculus.
Electrostatics:Coulomb’sLaw,ElectricFieldIntensity,ElectricFlux
Density,GaussLawandApplications,ElectricPotential,Maxwell’sTwo
EquationsforElectrostaticFields,IllustrativeProblems.Convection
andConductionCurrents,DielectricConstant,ContinuityEquation,
Capacitance,ExampleproblemsonRectangular,Cylindricaland
Sphericalcoordinates.
Magnetostatics:Biot-Sarvat'sLaw,Ampere’sCircuitalLawand
Applications,MagneticFluxDensity,TwoEquationsforMagnetostatic
Fields,MagneticScalarandVectorPotentials,ForcesduetoMagnetic
Fields,Ampere’sForceLaw,Inductance,Exampleproblemson
Rectangular,CylindricalandSphericalcoordinates.
Aditya Engineering College (A)
UNIT -III: ELECTROSTATICS & MAGNETOSTATICS (Syllabus )
Review on Co-ordinate Systems, Vector Calculus.
Electrostatics:Coulomb’sLaw,ElectricFieldIntensity,ElectricFlux
Density,GaussLawandApplications,ElectricPotential,Maxwell’sTwo
EquationsforElectrostaticFields,IllustrativeProblems.Convection
andConductionCurrents,DielectricConstant,ContinuityEquation,
Capacitance,ExampleproblemsonRectangular,Cylindricaland
Sphericalcoordinates.
Magnetostatics:Biot-Sarvat'sLaw,Ampere’sCircuitalLawand
Applications,MagneticFluxDensity,TwoEquationsforMagnetostatic
Fields,MagneticScalarandVectorPotentials,ForcesduetoMagnetic
Fields,Ampere’sForceLaw,Inductance,Exampleproblemson
Rectangular,CylindricalandSphericalcoordinates.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
Vector Algebra
Coordinate Systems
Vector Calculus
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
Vector Algebra
Coordinate Systems
Vector Calculus
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
VectorAlgebra:
•ScalarandVector
•UnitVector,PositionVectorandDistancevector
•VectorMultiplication(Dotproduct&CrossProduct)
•VectorAdditionandSubtraction
•Problems
CoordinateSystems:
CoordinateSystems:ThreecoordinatesystemsareusedtoanalyzeEM
waves.
1.CartesianCoordinates(x,y,z)
2.CylindricalCoordinates(ρ,ф,z)
3.SphericalCoordinates(r,θ,ф)
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
VectorAlgebra:
•ScalarandVector
•UnitVector,PositionVectorandDistancevector
•VectorMultiplication(Dotproduct&CrossProduct)
•VectorAdditionandSubtraction
•Problems
CoordinateSystems:
CoordinateSystems:ThreecoordinatesystemsareusedtoanalyzeEM
waves.
1.CartesianCoordinates(x,y,z)
2.CylindricalCoordinates(ρ,ф,z)
3.SphericalCoordinates(r,θ,ф)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
VectorCalculus:
•DifferentialLength,DifferentialAreaandDifferentialVolume
1.CartesianCoordinates
2.CylindricalCoordinates
3.SphericalCoordinates
•Line,SurfaceandVolumeIntegrals
•Del-Operator
1.GradientofaScalar
2.DivergenceofaVectorandDivergencetheorem
3.CurlofaVectorandStroke'stheorem
4.LaplacianofScalar
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
VectorCalculus:
•DifferentialLength,DifferentialAreaandDifferentialVolume
1.CartesianCoordinates
2.CylindricalCoordinates
3.SphericalCoordinates
•Line,SurfaceandVolumeIntegrals
•Del-Operator
1.GradientofaScalar
2.DivergenceofaVectorandDivergencetheorem
3.CurlofaVectorandStroke'stheorem
4.LaplacianofScalar
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
Aditya Engineering College (A)
Review of Coordinate Systems and Vector Calculus
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Vector Algebra
•Ascalarisaquantitythathasonlymagnitude.
•Avectorisquantitythathasbothmagnitudeanddirection.
•Avectorҧ�mayberepresentedas
ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
orsimplyas(�
�,�
�,�
�)
Where �
�
ᴧ
-Unit vector along �-axis
�
�
ᴧ
-Unit vector along �-axis
�
�
ᴧ
-Unit vector along �-axis
•ithasitsmagnitudethatrepresentedasҧ�,where
ҧ�=(�
�
�
+�
�
�
+�
�
�
)andtheunitvectorofഥ�is�
�
ᴧ
=
ഥ�
ҧ�
Aditya Engineering College (A)
Vector Algebra
•Ascalarisaquantitythathasonlymagnitude.
•Avectorisquantitythathasbothmagnitudeanddirection.
•Avectorҧ�mayberepresentedas
ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
orsimplyas(�
�,�
�,�
�)
Where �
�
ᴧ
-Unit vector along �-axis
�
�
ᴧ
-Unit vector along �-axis
�
�
ᴧ
-Unit vector along �-axis
•ithasitsmagnitudethatrepresentedasҧ�,where
ҧ�=(�
�
�
+�
�
�
+�
�
�
)andtheunitvectorofഥ�is�
�
ᴧ
=
ഥ�
ҧ�
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Vector cont.,
Aditya Engineering College (A)
Vector cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Vector Multiplication
•ScalarMultiplication(DOTPRODUCT)-�.�
•VectorMultiplication(CROSSPRODUCT)-�×�
DOTPRODUCT-�.�:
ҧ�.ത�=�����??????
��
Say ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
and
ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
then
ҧ�.ത�=�
�.�
�+�
�.�
�+�
�.�
�
CROSSPRODUCT-��:
ҧ�×ത�=���??????�??????
��??????
�
ᴧ
Say ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
and
ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
then
ҧ�×ത�=
Aditya Engineering College (A)
Vector Multiplication
•ScalarMultiplication(DOTPRODUCT)-�.�
•VectorMultiplication(CROSSPRODUCT)-�×�
DOTPRODUCT-�.�:
ҧ�.ത�=�����??????
��
Say ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
and
ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
then
ҧ�.ത�=�
�.�
�+�
�.�
�+�
�.�
�
CROSSPRODUCT-��:
ҧ�×ത�=���??????�??????
��??????
�
ᴧ
Say ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
and
ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
then
ҧ�×ത�=
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Position Vector and Distance Vector
ApointPincartesiancoordinate
systemmayberepresentedas(�,�,�).
Thepositionvector�
�(orradius
vector)ofpointPisthedirected
distancefromtheoriginOtoP.i.e,
�
�=��=�??????
�
ᴧ
+�??????
�
ᴧ
+�??????
�
ᴧ
Aditya Engineering College (A)
Position Vector and Distance Vector
ApointPincartesiancoordinate
systemmayberepresentedas(�,�,�).
Thepositionvector�
�(orradius
vector)ofpointPisthedirected
distancefromtheoriginOtoP.i.e,
�
�=��=�??????
�
ᴧ
+�??????
�
ᴧ
+�??????
�
ᴧ
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Position Vector and Distance Vector
Aditya Engineering College (A)
Position Vector and Distance Vector
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Position Vector and Distance Vector
Aditya Engineering College (A)
Position Vector and Distance Vector
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem 1
Aditya Engineering College (A)
Problem 1
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Vector addition and subtractction
Aditya Engineering College (A)
Vector addition and subtractction
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem 2 cont.,
Aditya Engineering College (A)
Problem 2 cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Cartesian Coordinates (x, y, z)
•A point P(�, �,�) is represented in
Cartesian coordinates as shown Fig.
•The range of the coordinate variables �, �,
and �are
−∞<�<∞
−∞<�<∞
−∞<�<∞
•�
�
ᴧ
-Unit vector along �-axis
•�
�
ᴧ
-Unit vector along �-axis
•�
�
ᴧ
-Unit vector along z-axis
Aditya Engineering College (A)
Cartesian Coordinates (x, y, z)
•A point P(�, �,�) is represented in
Cartesian coordinates as shown Fig.
•The range of the coordinate variables �, �,
and �are
−∞<�<∞
−∞<�<∞
−∞<�<∞
•�
�
ᴧ
-Unit vector along �-axis
•�
�
ᴧ
-Unit vector along �-axis
•�
�
ᴧ
-Unit vector along z-axis
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Cartesian Coordinates (x, y, z)
•Avectorҧ�representedinCartesiancoordinatesystemas
ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
orsimplyas(�
�,�
�,�
�)
Where �
�
ᴧ
-Unit vector along �-axis
�
�
ᴧ
-Unit vector along �-axis
�
�
ᴧ
-Unit vector along �-axis
•Ithasitsmagnitudethatrepresentedas�,where
�=(�
�
�
+�
�
�
+�
�
�
)andtheunitvectorofഥ�is�
�
ᴧ
=
ഥ�
�
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=1
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=0
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=0
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=0
�
�
ᴧ
�
�
ᴧ
= �
�
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
=0
�
�
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=−�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=−�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=−�
�
ᴧ
Aditya Engineering College (A)
Cartesian Coordinates (x, y, z)
•Avectorҧ�representedinCartesiancoordinatesystemas
ഥ�=�
�??????
�
ᴧ
+�
�??????
�
ᴧ
+�
�??????
�
ᴧ
orsimplyas(�
�,�
�,�
�)
Where �
�
ᴧ
-Unit vector along �-axis
�
�
ᴧ
-Unit vector along �-axis
�
�
ᴧ
-Unit vector along �-axis
•Ithasitsmagnitudethatrepresentedas�,where
�=(�
�
�
+�
�
�
+�
�
�
)andtheunitvectorofഥ�is�
�
ᴧ
=
ഥ�
�
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=1
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=0
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=0
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=0
�
�
ᴧ
�
�
ᴧ
= �
�
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
=0
�
�
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=−�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=−�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=−�
�
ᴧ
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Cylindrical Coordinates (ρ, ф, z)
•A point P(ρ, ф, z) is represented in
cylindrical coordinates as shown Fig.
•The range of the coordinate variables �,ф,
����are
0≤ρ<∞
0≤ф<2�
−∞<�<∞
•�
�
ᴧ
-Unit vector along �-axis
•�
∅
ᴧ
-Unit vector along ∅-axis
•�
�
ᴧ
-Unit vector along �-axis
Aditya Engineering College (A)
Cylindrical Coordinates (ρ, ф, z)
•A point P(ρ, ф, z) is represented in
cylindrical coordinates as shown Fig.
•The range of the coordinate variables �,ф,
����are
0≤ρ<∞
0≤ф<2�
−∞<�<∞
•�
�
ᴧ
-Unit vector along �-axis
•�
∅
ᴧ
-Unit vector along ∅-axis
•�
�
ᴧ
-Unit vector along �-axis
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Cylindrical Coordinates (ρ, ф, z)cont.,
•Avectorҧ�representedinCylindricalcoordinatesystemas
ഥ�=�
�??????
�
ᴧ
+�
∅??????
∅
ᴧ
+�
�??????
�
ᴧ
orsimplyas(�
�,�
∅,�
�)
•Where�
�
ᴧ
-Unit vector along �-axis
�
∅
ᴧ
-Unit vector along ∅-axis
�
�
ᴧ
-Unit vector along �-axis
•Ithasitsmagnitudethatrepresentedas�,where
�=(�
�
�
+�
∅
�
+�
�
�
)andtheunitvectorofഥ�is�
�
ᴧ
=
ഥ�
�
�
�
ᴧ
.�
�
ᴧ
=�
∅
ᴧ
.�
∅
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=1
�
�
ᴧ
.�
∅
ᴧ
=�
∅
ᴧ
.�
�
ᴧ
=0
�
∅
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
∅
ᴧ
=0
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=0
�
�
ᴧ
�
�
ᴧ
= �
∅
ᴧ
�
∅
ᴧ
=�
�
ᴧ
�
�
ᴧ
=0
�
�
ᴧ
�
∅
ᴧ
=�
�
ᴧ
�
∅
ᴧ
�
�
ᴧ
=−�
�
ᴧ
�
∅
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
�
∅
ᴧ
=−�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=�
∅
ᴧ
�
�
ᴧ
�
�
ᴧ
=−�
∅
ᴧ
Aditya Engineering College (A)
Cylindrical Coordinates (ρ, ф, z)cont.,
•Avectorҧ�representedinCylindricalcoordinatesystemas
ഥ�=�
�??????
�
ᴧ
+�
∅??????
∅
ᴧ
+�
�??????
�
ᴧ
orsimplyas(�
�,�
∅,�
�)
•Where�
�
ᴧ
-Unit vector along �-axis
�
∅
ᴧ
-Unit vector along ∅-axis
�
�
ᴧ
-Unit vector along �-axis
•Ithasitsmagnitudethatrepresentedas�,where
�=(�
�
�
+�
∅
�
+�
�
�
)andtheunitvectorofഥ�is�
�
ᴧ
=
ഥ�
�
�
�
ᴧ
.�
�
ᴧ
=�
∅
ᴧ
.�
∅
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=1
�
�
ᴧ
.�
∅
ᴧ
=�
∅
ᴧ
.�
�
ᴧ
=0
�
∅
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
∅
ᴧ
=0
�
�
ᴧ
.�
�
ᴧ
=�
�
ᴧ
.�
�
ᴧ
=0
�
�
ᴧ
�
�
ᴧ
= �
∅
ᴧ
�
∅
ᴧ
=�
�
ᴧ
�
�
ᴧ
=0
�
�
ᴧ
�
∅
ᴧ
=�
�
ᴧ
�
∅
ᴧ
�
�
ᴧ
=−�
�
ᴧ
�
∅
ᴧ
�
�
ᴧ
=�
�
ᴧ
�
�
ᴧ
�
∅
ᴧ
=−�
�
ᴧ
�
�
ᴧ
�
�
ᴧ
=�
∅
ᴧ
�
�
ᴧ
�
�
ᴧ
=−�
∅
ᴧ
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Spherical Coordinates (�,??????,∅)
•A point P(ρ, ф, z) is represented Spherical
coordinates as shown Fig.
•The range of the coordinate variables �,ф,
����are
0≤�<∞
0≤∅<2�
0≤θ<�
•�
??????
ᴧ
-Unit vector along �-axis
•�
∅
ᴧ
-Unit vector along ∅-axis
•�
??????
ᴧ
-Unit vector along ??????-axis
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Spherical Coordinates (�,??????,∅)
•A point P(ρ, ф, z) is represented Spherical
coordinates as shown Fig.
•The range of the coordinate variables �,ф,
����are
0≤�<∞
0≤∅<2�
0≤θ<�
•�
??????
ᴧ
-Unit vector along �-axis
•�
∅
ᴧ
-Unit vector along ∅-axis
•�
??????
ᴧ
-Unit vector along ??????-axis
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Spherical Coordinates (�,??????,∅)cont.,
•Avectorҧ�representedinsphericalcoordinatesystemas
ഥ�=�
�??????
�
ᴧ
+�
????????????
??????
ᴧ
+�
∅??????
∅
ᴧ
orsimplyas(�
�,�
??????,�
∅)
•Where�
??????
ᴧ
-Unit vector along �-axis
�
∅
ᴧ
-Unit vector along ∅-axis
�
??????
ᴧ
-Unit vector along ??????-axis
•Ithasitsmagnitudethatrepresentedas�,where
�=(�
�
�
+�
??????
�
+�
∅
�
)andtheunitvectorofഥ�is�
�
ᴧ
=
ഥ�
�
�
??????
ᴧ
.�
??????
ᴧ
=�
??????
ᴧ
.�
??????
ᴧ
=�
∅
ᴧ
.�
∅
ᴧ
=1
�
??????
ᴧ
.�
??????
ᴧ
=�
??????
ᴧ
.�
??????
ᴧ
=0
�
??????
ᴧ
.�
∅
ᴧ
=�
∅
ᴧ
.�
??????
ᴧ
=0
�
∅
ᴧ
.�
??????
ᴧ
=�
??????
ᴧ
.�
∅
ᴧ
=0
�
??????
ᴧ
�
??????
ᴧ
= �
??????
ᴧ
�
??????
ᴧ
=�
∅
ᴧ
�
∅
ᴧ
=0
�
??????
ᴧ
�
??????
ᴧ
=�
∅
ᴧ
�
??????
ᴧ
�
??????
ᴧ
=−�
∅
ᴧ
�
??????
ᴧ
�
∅
ᴧ
=�
??????
ᴧ
�
∅
ᴧ
�
??????
ᴧ
=−�
??????
ᴧ
�
∅
ᴧ
�
??????
ᴧ
=�
??????
ᴧ
�
??????
ᴧ
�
∅
ᴧ
=−�
??????
ᴧ
Aditya Engineering College (A)
Spherical Coordinates (�,??????,∅)cont.,
•Avectorҧ�representedinsphericalcoordinatesystemas
ഥ�=�
�??????
�
ᴧ
+�
????????????
??????
ᴧ
+�
∅??????
∅
ᴧ
orsimplyas(�
�,�
??????,�
∅)
•Where�
??????
ᴧ
-Unit vector along �-axis
�
∅
ᴧ
-Unit vector along ∅-axis
�
??????
ᴧ
-Unit vector along ??????-axis
•Ithasitsmagnitudethatrepresentedas�,where
�=(�
�
�
+�
??????
�
+�
∅
�
)andtheunitvectorofഥ�is�
�
ᴧ
=
ഥ�
�
�
??????
ᴧ
.�
??????
ᴧ
=�
??????
ᴧ
.�
??????
ᴧ
=�
∅
ᴧ
.�
∅
ᴧ
=1
�
??????
ᴧ
.�
??????
ᴧ
=�
??????
ᴧ
.�
??????
ᴧ
=0
�
??????
ᴧ
.�
∅
ᴧ
=�
∅
ᴧ
.�
??????
ᴧ
=0
�
∅
ᴧ
.�
??????
ᴧ
=�
??????
ᴧ
.�
∅
ᴧ
=0
�
??????
ᴧ
�
??????
ᴧ
= �
??????
ᴧ
�
??????
ᴧ
=�
∅
ᴧ
�
∅
ᴧ
=0
�
??????
ᴧ
�
??????
ᴧ
=�
∅
ᴧ
�
??????
ᴧ
�
??????
ᴧ
=−�
∅
ᴧ
�
??????
ᴧ
�
∅
ᴧ
=�
??????
ᴧ
�
∅
ᴧ
�
??????
ᴧ
=−�
??????
ᴧ
�
∅
ᴧ
�
??????
ᴧ
=�
??????
ᴧ
�
??????
ᴧ
�
∅
ᴧ
=−�
??????
ᴧ
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coordinate transformation (�,�,�)↔(ρ, ф, z)
•TherelationshipbetweenCartesiancoordinatesandCylindrical
CoordinatesillustratedintheFig.below.
(�,�,�)→(ρ,ф,z)
•(ρ,ф,z)→(�,�,�)
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Coordinate transformation (�,�,�)↔(ρ, ф, z)
•TherelationshipbetweenCartesiancoordinatesandCylindrical
CoordinatesillustratedintheFig.below.
(�,�,�)→(ρ,ф,z)
•(ρ,ф,z)→(�,�,�)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coordinate transformation (�,�,�)↔(�,??????,∅)
•TherelationshipbetweenCylindricalCoordinatesandSpherical
coordinatesillustratedintheFig.below.
(�,�,�)→(�,??????,∅)
(�,??????,∅)→(�,�,�)
Aditya Engineering College (A)
Coordinate transformation (�,�,�)↔(�,??????,∅)
•TherelationshipbetweenCylindricalCoordinatesandSpherical
coordinatesillustratedintheFig.below.
(�,�,�)→(�,??????,∅)
(�,??????,∅)→(�,�,�)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
VECTOR CALCULUS
•DifferentialLength,DifferentialAreaandDifferentialVolume
1.CartesianCoordinates
2.CylindricalCoordinates
3.SphericalCoordinates
•Line,SurfaceandVolumeIntegrals
•Del-Operator
1.GradientofaScalar
2.DivergenceofaVectorandDivergencetheorem
3.CurlofaVectorandStroke'stheorem
4.LaplacianofScalar
Aditya Engineering College (A)
VECTOR CALCULUS
•DifferentialLength,DifferentialAreaandDifferentialVolume
1.CartesianCoordinates
2.CylindricalCoordinates
3.SphericalCoordinates
•Line,SurfaceandVolumeIntegrals
•Del-Operator
1.GradientofaScalar
2.DivergenceofaVectorandDivergencetheorem
3.CurlofaVectorandStroke'stheorem
4.LaplacianofScalar
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Differential Length, Differential Area and Differential Volume
(Cartesian Coordinates)
Aditya Engineering College (A)
Differential Length, Differential Area and Differential Volume
(Cartesian Coordinates)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Differential Length, Differential Area and Differential Volume
(Cylindrical Coordinates)
Aditya Engineering College (A)
Differential Length, Differential Area and Differential Volume
(Cylindrical Coordinates)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Differential Length, Differential Area and Differential Volume
(Spherical Coordinates)
Aditya Engineering College (A)
Differential Length, Differential Area and Differential Volume
(Spherical Coordinates)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Line , Surface and Volume Integrals
LineIntegral:
Thelineintegralofavectorҧ�isdefinedastheintegrationoftangentialcomponentofa
vectorthatvectoralongacurveL.i.e.,
??????
ҧ�.��=
�
�
ҧ����??????��Thisislineintegralofҧ�aroundL.
IfpathoftheintegralisclosedcurvesuchasabcbaasinFig.,theaboveequation
becomesclosedcontourintegralas
ׯ
??????
ҧ�.��=ׯ
??????
ҧ����??????��TheclosedlineintegraliscalledCirculationofഥ�.
Aditya Engineering College (A)
Line , Surface and Volume Integrals
LineIntegral:
Thelineintegralofavectorҧ�isdefinedastheintegrationoftangentialcomponentofa
vectorthatvectoralongacurveL.i.e.,
??????
ҧ�.��=
�
�
ҧ����??????��Thisislineintegralofҧ�aroundL.
IfpathoftheintegralisclosedcurvesuchasabcbaasinFig.,theaboveequation
becomesclosedcontourintegralas
ׯ
??????
ҧ�.��=ׯ
??????
ҧ����??????��TheclosedlineintegraliscalledCirculationofഥ�.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Line , Surface and Volume Integrals cont.,
SurfaceIntegral:
Thesurfaceintegralofavectorҧ�throughsurfaceSisdefinedasfluxofഥ�.
o
Thesurfaceintegralofavectorҧ�throughaclosedsurfaceisdefinedasthenetoutward
fluxofഥ�.i.e.
VolumeIntegral:
Thevolumeintegralofascalar�
�isdefinedas
�
�
���
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Line , Surface and Volume Integrals cont.,
SurfaceIntegral:
Thesurfaceintegralofavectorҧ�throughsurfaceSisdefinedasfluxofഥ�.
o
Thesurfaceintegralofavectorҧ�throughaclosedsurfaceisdefinedasthenetoutward
fluxofഥ�.i.e.
VolumeIntegral:
Thevolumeintegralofascalar�
�isdefinedas
�
�
���
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Del-Operator (�)
•Thedeloperatorwrittenas??????,isthevectordifferentialoperator.Incartesian
coordinates,cylindricalcoordinatesandsphericalcoordinates,itisrepresentedas
•Thevectordifferentialoperatorisalsoknownasgradientoperatoranditisnot
vectoritself.
•Theoperatorisusefulindefining
1.TheGradientofaScalarV,writtenas??????�
2.TheDivergenceofaVectorҧ�,writtenas??????.ҧ�
3.TheCurlofaVectorҧ�,writtenas??????×ҧ�
4.TheLaplacianofaScalarV,writtenas??????
2
�
Aditya Engineering College (A)
Del-Operator (�)
•Thedeloperatorwrittenas??????,isthevectordifferentialoperator.Incartesian
coordinates,cylindricalcoordinatesandsphericalcoordinates,itisrepresentedas
•Thevectordifferentialoperatorisalsoknownasgradientoperatoranditisnot
vectoritself.
•Theoperatorisusefulindefining
1.TheGradientofaScalarV,writtenas??????�
2.TheDivergenceofaVectorҧ�,writtenas??????.ҧ�
3.TheCurlofaVectorҧ�,writtenas??????×ҧ�
4.TheLaplacianofaScalarV,writtenas??????
2
�
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
The Gradient of a Scalar V, written as ∇??????
•ThegradientofascalarfieldVisavectorthatrepresentsboththemagnitudeandthe
directionofthemaximumspacerateofincreaseofV.writtenas??????,isthevector
differentialoperator.
Aditya Engineering College (A)
The Gradient of a Scalar V, written as ∇??????
•ThegradientofascalarfieldVisavectorthatrepresentsboththemagnitudeandthe
directionofthemaximumspacerateofincreaseofV.writtenas??????,isthevector
differentialoperator.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
The Divergence of a Vector ഥ�, written as �.ഥ�
•Definition:Thedivergenceofavectorfieldҧ�atapointPistheoutwardfluxperunit
volumeasthevolumeshrinksaboutP.
•DivergenceTheorem:Thedivergencetheoremstatesthatthetotaloutwardfluxofa
vectorfieldҧ�throughtheclosedsurfaceSisthesaneasthevolumeintegralofthe
divergenceofҧ�.
Aditya Engineering College (A)
The Divergence of a Vector ഥ�, written as �.ഥ�
•Definition:Thedivergenceofavectorfieldҧ�atapointPistheoutwardfluxperunit
volumeasthevolumeshrinksaboutP.
•DivergenceTheorem:Thedivergencetheoremstatesthatthetotaloutwardfluxofa
vectorfieldҧ�throughtheclosedsurfaceSisthesaneasthevolumeintegralofthe
divergenceofҧ�.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
The Laplacian of a Scalar V, written as ??????
�
�
•Definition:TheLaplacianofascalarfield�,writtenas??????
�
�,isthedivergenceofthe
gradientof�.
??????.�.�??????��??????�????????????����=??????.??????�=??????
�
�
Aditya Engineering College (A)
The Laplacian of a Scalar V, written as ??????
�
�
•Definition:TheLaplacianofascalarfield�,writtenas??????
�
�,isthedivergenceofthe
gradientof�.
??????.�.�??????��??????�????????????����=??????.??????�=??????
�
�
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
The Curl of a Vector ഥ�, written as �×ഥ�
•Definition:Thecurlofavectorfieldҧ�isdefiedasthemaximumcirculationofҧ�per
unitareaastheareatendstozeroandwhosedirectionisthenormaldirectionofthe
area.
•Stoke’sTheorem:TheStoke’stheoremstatesthatthecirculationofavectorfieldҧ�
throughtheclosedpathListhesaneasthesurfaceintegralofthecurlofҧ�.
Aditya Engineering College (A)
The Curl of a Vector ഥ�, written as �×ഥ�
•Definition:Thecurlofavectorfieldҧ�isdefiedasthemaximumcirculationofҧ�per
unitareaastheareatendstozeroandwhosedirectionisthenormaldirectionofthe
area.
•Stoke’sTheorem:TheStoke’stheoremstatesthatthecirculationofavectorfieldҧ�
throughtheclosedpathListhesaneasthesurfaceintegralofthecurlofҧ�.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
FORMULAS
Aditya Engineering College (A)
FORMULAS
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Overview on Unit-3, Unit-4 and Unit-5
Aditya Engineering College (A)
Overview on Unit-3, Unit-4 and Unit-5
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Overview on Unit-3, Unit-4 and Unit-5
Aditya Engineering College (A)
Overview on Unit-3, Unit-4 and Unit-5
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
ELECTROSTATICS -Study of Static Electric Field
Electrostatics–FreeSpace
•Coulomb'slaw
•ElectricFieldIntensity(E)
•Chargedistributions–Line,SurfaceandVolumeCharge
•ElectricFluxDensity(D)
•Gauss’slaw-1
st
Maxwell’sEquation
•Electripotential(V)
•RelationbetweenEandV-2
nd
Maxwell’sEquation
Aditya Engineering College (A)
ELECTROSTATICS -Study of Static Electric Field
Electrostatics–FreeSpace
•Coulomb'slaw
•ElectricFieldIntensity(E)
•Chargedistributions–Line,SurfaceandVolumeCharge
•ElectricFluxDensity(D)
•Gauss’slaw-1
st
Maxwell’sEquation
•Electripotential(V)
•RelationbetweenEandV-2
nd
Maxwell’sEquation
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
ELECTROSTATICSCont.,
Electrostatics–MaterialSpace
•ConvectionandConductionCurrents
•DielectricConstant
•ContinuityEquation
•Capacitance
•ExampleproblemsonRectangular,CylindricalandSpherical
coordinates.
Aditya Engineering College (A)
ELECTROSTATICSCont.,
Electrostatics–MaterialSpace
•ConvectionandConductionCurrents
•DielectricConstant
•ContinuityEquation
•Capacitance
•ExampleproblemsonRectangular,CylindricalandSpherical
coordinates.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law
Coulomb's law states that the force between two point charges �
1and �
2is :
1.Along the line joining them
2.Directly proportional to the product �
1�
2of the charges.
3.Inversely proportional to the square of the distance �between them.
Expressedmathematicallyas�∝
����
�
�
=>�=�
����
�
�
Wherekistheproportionalconstanti.e�=
�
��??????�
Where??????
0isthepermittivityofthefreespaceandhavethevalue
??????
�=�.�����
−��
=
��
−�
���
��
Aditya Engineering College (A)
Coulomb's law
Coulomb's law states that the force between two point charges �
1and �
2is :
1.Along the line joining them
2.Directly proportional to the product �
1�
2of the charges.
3.Inversely proportional to the square of the distance �between them.
Expressedmathematicallyas�∝
����
�
�
=>�=�
����
�
�
Wherekistheproportionalconstanti.e�=
�
��??????�
Where??????
0isthepermittivityofthefreespaceandhavethevalue
??????
�=�.�����
−��
=
��
−�
���
��
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Coulomb's law cont.,
Aditya Engineering College (A)
Coulomb's law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
FORMULUS
•The force acting on a point charge �at point �due to another point charge
�
1at �
1is give by
�=
�
1Q(r−�
�)
4�??????
??????r−�
�
3
•The force acting on a point charge �at point �due to several point charges
point charge �
1,�
2,�
3,…….,�
??????at �
1,�
2,�
3…….�
??????, is give by
�=
�
4�??????
??????
??????=1
??????
�
??????(r−�
??????)
r−�
??????
3
Aditya Engineering College (A)
FORMULUS
•The force acting on a point charge �at point �due to another point charge
�
1at �
1is give by
�=
�
1Q(r−�
�)
4�??????
??????r−�
�
3
•The force acting on a point charge �at point �due to several point charges
point charge �
1,�
2,�
3,…….,�
??????at �
1,�
2,�
3…….�
??????, is give by
�=
�
4�??????
??????
??????=1
??????
�
??????(r−�
??????)
r−�
??????
3
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric field Intensity (E)
Aditya Engineering College (A)
Electric field Intensity (E)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric field Intensity (E)
Aditya Engineering College (A)
Electric field Intensity (E)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
FORMULUS
•The electric field intensity at point �due to another point charge �
1at �
1is
give by
E=
�
�(�−�
�)
4�??????
��−�
�
3
•The electric field intensity at point �due to several point charges point
charge �
1,�
2, �
3,……., �
??????at �
1,�
2,�
3…….�
??????, is give by
E=
1
4�??????
??????
??????=1
??????
�
??????(r−�
??????)
r−�
??????
3
Aditya Engineering College (A)
FORMULUS
•The electric field intensity at point �due to another point charge �
1at �
1is
give by
E=
�
�(�−�
�)
4�??????
��−�
�
3
•The electric field intensity at point �due to several point charges point
charge �
1,�
2, �
3,……., �
??????at �
1,�
2,�
3…….�
??????, is give by
E=
1
4�??????
??????
??????=1
??????
�
??????(r−�
??????)
r−�
??????
3
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-1
Point charges 1mC and -2mC are located at (3,2,-1) and (-1,-1,4), respectively.
Calculate the electric force on a 10nC charge located at (0,3,1) and the electric
field intensity at that point.
Aditya Engineering College (A)
Example-1
Point charges 1mC and -2mC are located at (3,2,-1) and (-1,-1,4), respectively.
Calculate the electric force on a 10nC charge located at (0,3,1) and the electric
field intensity at that point.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-1
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Example-1
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-1
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Example-1
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-2
Point charges 5nC and -2nC are located at (2,0,4) and (-3,0,5), respectively.
(a)Determine the force on a 1nC point charge located at (1,-3,7).
(b)Find the electric field E at (1,-3,7).
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Example-2
Point charges 5nC and -2nC are located at (2,0,4) and (-3,0,5), respectively.
(a)Determine the force on a 1nC point charge located at (1,-3,7).
(b)Find the electric field E at (1,-3,7).
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-1
Point charges 1mC and -2mC are located at (3,2,-1) and (-1,-1,4), respectively.
Calculate the electric force on a 10nC charge located at (0,3,1) and the electric
field intensity at that point.
Aditya Engineering College (A)
Example-1
Point charges 1mC and -2mC are located at (3,2,-1) and (-1,-1,4), respectively.
Calculate the electric force on a 10nC charge located at (0,3,1) and the electric
field intensity at that point.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-1
Aditya Engineering College (A)
Example-1
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-1
Aditya Engineering College (A)
Example-1
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-2
Point charges 5nC and -2nC are located at (2,0,4) and (-3,0,5), respectively.
(a)Determine the force on a 1nC point charge located at (1,-3,7).
(b)Find the electric field E at (1,-3,7).
Aditya Engineering College (A)
Example-2
Point charges 5nC and -2nC are located at (2,0,4) and (-3,0,5), respectively.
(a)Determine the force on a 1nC point charge located at (1,-3,7).
(b)Find the electric field E at (1,-3,7).
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-2
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Example-2
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-2
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Example-2
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Charge Distributions
There are four possible charge distributions:
1.Point Charge -Q
2.Line charge distribution with line charge density �
??????(C/m)
3.Surface charge distribution with surfacecharge density �
??????(C/�
2
)
4.Volume charge distribution with volumecharge density �
�(C/�
3
)
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Charge Distributions
There are four possible charge distributions:
1.Point Charge -Q
2.Line charge distribution with line charge density �
??????(C/m)
3.Surface charge distribution with surfacecharge density �
??????(C/�
2
)
4.Volume charge distribution with volumecharge density �
�(C/�
3
)
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Charge Distributions
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Charge Distributions
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
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Electric field due to Line Charge Distribution
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
Aditya Engineering College (A)
Electric field due to Line Charge Distribution
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem-3
Calculate E at (1,1,-1) due to the line �=0,�=2caries charge 10�nC/m.
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Problem-3
Calculate E at (1,1,-1) due to the line �=0,�=2caries charge 10�nC/m.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem-3
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Problem-3
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem-4
Calculate E at (4,0,0) due to the line charge 3�mC/m along the y-axis.
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Problem-4
Calculate E at (4,0,0) due to the line charge 3�mC/m along the y-axis.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Evaluation of Electric field intensity due to
surface Charge Distribution
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Evaluation of Electric field intensity due to
surface Charge Distribution
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Evaluation of Electric field intensity due to
surface Charge Distribution cont.,
Aditya Engineering College (A)
Evaluation of Electric field intensity due to
surface Charge Distribution cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Evaluation of Electric field intensity due to
surface Charge Distribution cont.,
Aditya Engineering College (A)
Evaluation of Electric field intensity due to
surface Charge Distribution cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Evaluation of Electric field intensity due to
surface Charge Distribution cont.,
Aditya Engineering College (A)
Evaluation of Electric field intensity due to
surface Charge Distribution cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem-5
Find electric field intensity E at (2,1,3) due to THE plane �=1carry
charges 10��/�
2
.
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Problem-5
Find electric field intensity E at (2,1,3) due to THE plane �=1carry
charges 10��/�
2
.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem-6
Find electric field intensity E at (1,1,-1) due to two planes �=2����=3
carry charges 10��/�
2
and 15��/�
2
respectively.
Aditya Engineering College (A)
Problem-6
Find electric field intensity E at (1,1,-1) due to two planes �=2����=3
carry charges 10��/�
2
and 15��/�
2
respectively.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem-6 cont.,
Aditya Engineering College (A)
Problem-6 cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric flux density(D)
•The electric flux per unit area is called Electric flux density D.
•The electric flux is expressed in terms of flux density as
??????=න
�
ഥ�.��
•It is independent of permittivity of the media.
•It can also be written expressed in terms of field intensity as
ഥ�=∈
�
ഥ�
•It is also called Electric displacement.
•It measured in �/�
2
.
Aditya Engineering College (A)
Electric flux density(D)
•The electric flux per unit area is called Electric flux density D.
•The electric flux is expressed in terms of flux density as
??????=න
�
ഥ�.��
•It is independent of permittivity of the media.
•It can also be written expressed in terms of field intensity as
ഥ�=∈
�
ഥ�
•It is also called Electric displacement.
•It measured in �/�
2
.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric flux density(D)
Aditya Engineering College (A)
Electric flux density(D)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem-7
Find D at (4,0,3) if there is a point charge -5πmCat (2,-3,4) and a line
charge 3π mC/m along the line x=-2 and z=3.
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Problem-7
Find D at (4,0,3) if there is a point charge -5πmCat (2,-3,4) and a line
charge 3π mC/m along the line x=-2 and z=3.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Problem-7 cont.,
Find D at (4,0,3) if there is a point charge -5πmCat (2,-3,4) and a line
charge 3π mC/m along the line x=-2 and z=3.
Aditya Engineering College (A)
Problem-7 cont.,
Find D at (4,0,3) if there is a point charge -5πmCat (2,-3,4) and a line
charge 3π mC/m along the line x=-2 and z=3.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Gauss’s law –Maxwell’s 1
st
Equation
Statement:Gauss’s law states that the total electric flux ??????through any closed
surface is equal to the total charge enclosed by that surface.
Thus ??????
������=�
��������
-----------(1)
that is ??????
������=ׯ
�
�.��---------(2)
But the total charge enclosed by that surface �
��������=
�
�
���--------(3)
From equations (2) & (3), we have
∴ׯ
�
�.��=
�
�
���------------(4)
From Divergence theorem ׯ
�
�.��=
�
??????.���-------------(5)
From equations (4) &(5), we have ∴
�
�.���=
�
�
���or �.�=�
�
This is 1
st
Maxwell’s Equation for STATIC ELECTIC FIELD.
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Gauss’s law –Maxwell’s 1
st
Equation
Statement:Gauss’s law states that the total electric flux ??????through any closed
surface is equal to the total charge enclosed by that surface.
Thus ??????
������=�
��������
-----------(1)
that is ??????
������=ׯ
�
�.��---------(2)
But the total charge enclosed by that surface �
��������=
�
�
���--------(3)
From equations (2) & (3), we have
∴ׯ
�
�.��=
�
�
���------------(4)
From Divergence theorem ׯ
�
�.��=
�
??????.���-------------(5)
From equations (4) &(5), we have ∴
�
�.���=
�
�
���or �.�=�
�
This is 1
st
Maxwell’s Equation for STATIC ELECTIC FIELD.
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric Potential(V)
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Electric Potential(V)
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric Potential(V) cont.,
Aditya Engineering College (A)
Electric Potential(V) cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric Potential(V) cont.,
Aditya Engineering College (A)
Electric Potential(V) cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric Potential(V) cont.,
Aditya Engineering College (A)
Electric Potential(V) cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric Potential(V) cont.,
Aditya Engineering College (A)
Electric Potential(V) cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric Potential(V)-2
nd
Maxwell’s Equation
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Electric Potential(V)-2
nd
Maxwell’s Equation
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Electric Potential(V)-2
nd
Maxwell’s Equation cont.,
Aditya Engineering College (A)
Electric Potential(V)-2
nd
Maxwell’s Equation cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Relation between ഥ�??????���
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Relation between ഥ�??????���
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Relation between ഥ�??????���cont.,
Aditya Engineering College (A)
Relation between ഥ�??????���cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
POLARIZATION -Process
•When apply an electric field to dielectric material, then the positive charges
and negative charges are displaced from its equilibrium position and the
dipoles are generated as shown in figure.
•The dipole is formed by two equal amount of charges separated by a small
distance as shown in figure above.
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POLARIZATION -Process
•When apply an electric field to dielectric material, then the positive charges
and negative charges are displaced from its equilibrium position and the
dipoles are generated as shown in figure.
•The dipole is formed by two equal amount of charges separated by a small
distance as shown in figure above.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
POLARIZATION ഥ�
•The process of generating dipoles is called Polarization.
•The dipole moment of a dipole is give by ҧ�=�ҧ�
•The measure of intensity of polarization is called polarization P which is
proportional to the electric flux density D as
�∝�=>�∝??????
0�
=>�=χ
�??????
0�
Where χ
�is called electric susceptibility of the material
??????
0is the permittivity I free space
�is the applied electric field
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POLARIZATION ഥ�
•The process of generating dipoles is called Polarization.
•The dipole moment of a dipole is give by ҧ�=�ҧ�
•The measure of intensity of polarization is called polarization P which is
proportional to the electric flux density D as
�∝�=>�∝??????
0�
=>�=χ
�??????
0�
Where χ
�is called electric susceptibility of the material
??????
0is the permittivity I free space
�is the applied electric field
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
DILECTRIC CONSTANT (??????
�)
•The total electric flux density D , in a dielectric material is
�=??????�………(1)
Which is equal to the sum of flux density in free space and polarization ത�D=
??????
0�+ത�
=??????
0�+χ
�??????
0�(Since ത�=χ
�??????
0�)
=??????
01+χ
��
=??????
0??????
??????�…………(2)
Where ??????
??????=1+χ
�is called Dielectric constant.
∴∈�
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DILECTRIC CONSTANT (??????
�)
•The total electric flux density D , in a dielectric material is
�=??????�………(1)
Which is equal to the sum of flux density in free space and polarization ത�D=
??????
0�+ത�
=??????
0�+χ
�??????
0�(Since ത�=χ
�??????
0�)
=??????
01+χ
��
=??????
0??????
??????�…………(2)
Where ??????
??????=1+χ
�is called Dielectric constant.
∴∈�
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
DILECTRIC CONSTANT (??????
�) cont.,
From equation (1) and (2) we can write �=??????�=??????
0??????
??????�
∴??????=??????
0??????
??????=>??????
??????=
??????
??????
0
Dielectric constant(Relative permittivity):It is the ration of the permittivity
of the dielectric material to the permittivity of free space.
Dielectric field strength:It is the maximum electric field that a dielectric
material can tolerate or withstand without electrical breakdown.
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DILECTRIC CONSTANT (??????
�) cont.,
From equation (1) and (2) we can write �=??????�=??????
0??????
??????�
∴??????=??????
0??????
??????=>??????
??????=
??????
??????
0
Dielectric constant(Relative permittivity):It is the ration of the permittivity
of the dielectric material to the permittivity of free space.
Dielectric field strength:It is the maximum electric field that a dielectric
material can tolerate or withstand without electrical breakdown.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Convection and Conduction Currents
•Current: The current flowing through a given area is the electric charge
passing through the area per unit time.
That is I=
�??????
��
=
∆??????
∆�
………….(1)
•Current density: The current density is the flow of current per unit area.
That is J=
∆??????
∆??????
=>∆�=�∆�
=>∆�=�.∆�
Therefore the total current flowing through a surface S is
�=
??????
�.��……….(2)
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Convection and Conduction Currents
•Current: The current flowing through a given area is the electric charge
passing through the area per unit time.
That is I=
�??????
��
=
∆??????
∆�
………….(1)
•Current density: The current density is the flow of current per unit area.
That is J=
∆??????
∆??????
=>∆�=�∆�
=>∆�=�.∆�
Therefore the total current flowing through a surface S is
�=
??????
�.��……….(2)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Convection and Conduction Currents cont.,
Convection Current:
•Convectioncurrent,asdifferentfromconductioncurrent,doesnot
involveconductorsandconsequentlydoesnotsatisfyOhm’slaw.
•Itoccurswhencurrentflowthroughaninsulatingmaterialsuchas
liquid,rarefiedgas,oravacuum.Abeamofelectronsinavacuumtube
isanexampleforconvectioncurrent.
•ConsiderafilamentasshowninFig.below.
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Convection and Conduction Currents cont.,
Convection Current:
•Convectioncurrent,asdifferentfromconductioncurrent,doesnot
involveconductorsandconsequentlydoesnotsatisfyOhm’slaw.
•Itoccurswhencurrentflowthroughaninsulatingmaterialsuchas
liquid,rarefiedgas,oravacuum.Abeamofelectronsinavacuumtube
isanexampleforconvectioncurrent.
•ConsiderafilamentasshowninFig.below.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Convection and Conduction Currents cont.,
•If there is a flow of charge , of density �
�, at velocity �=�
�??????
�, from
equation (1) , the current through the filament is
∆�=
∆??????
∆�
=�
�∆�
∆�
∆�
=�
�∆��
�……………(3)
•The current density in y-direction is given by
�
�=
∆�
∆�
=>�
�=�
��
�
Hence in general
�=�
�??????……………(4)
The current Iis called convection current and the Jis called the convection
current density.
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Convection and Conduction Currents cont.,
•If there is a flow of charge , of density �
�, at velocity �=�
�??????
�, from
equation (1) , the current through the filament is
∆�=
∆??????
∆�
=�
�∆�
∆�
∆�
=�
�∆��
�……………(3)
•The current density in y-direction is given by
�
�=
∆�
∆�
=>�
�=�
��
�
Hence in general
�=�
�??????……………(4)
The current Iis called convection current and the Jis called the convection
current density.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Convection and Conduction Currents cont.,
Conduction Current:
•Conductioncurrentrequiresaconductor.Aconductorischaracterizedbya
largenumberoffreeelectronsthatprovideconductioncurrentduetoan
impressedelectricfield.WhenanelectricfieldEisapplied,theforceactingon
anelectronwithcharge−eis
F=−eE.........(1)
•ButfromNewton'slawtheforceactingonelectriccharge−e,movingwith
velocity�is
�=
??????�
??????
……….(2)
where??????istheaveragetimebetweencollisionsand�isthemassofthe−e.
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Convection and Conduction Currents cont.,
Conduction Current:
•Conductioncurrentrequiresaconductor.Aconductorischaracterizedbya
largenumberoffreeelectronsthatprovideconductioncurrentduetoan
impressedelectricfield.WhenanelectricfieldEisapplied,theforceactingon
anelectronwithcharge−eis
F=−eE.........(1)
•ButfromNewton'slawtheforceactingonelectriccharge−e,movingwith
velocity�is
�=
??????�
??????
……….(2)
where??????istheaveragetimebetweencollisionsand�isthemassofthe−e.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Convection and Conduction Currents cont.,
Fromequations(1)and(2)
??????�
??????
=−eE
=>�=−
e??????
??????
E………….(3)
•Ifthereare�electronsperunitvolume,thenthevolumechargedensityis
givenby�
�=−��…………(4)
•Thustheconductioncurrentdensityis
�=�
�??????=
��
�
??????
�
�=??????�
or �=??????�…………(5)ItisthepointformofOhm’slaw.
Where??????=
��
�
??????
�
istheconductivityoftheconductor.
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Convection and Conduction Currents cont.,
Fromequations(1)and(2)
??????�
??????
=−eE
=>�=−
e??????
??????
E………….(3)
•Ifthereare�electronsperunitvolume,thenthevolumechargedensityis
givenby�
�=−��…………(4)
•Thustheconductioncurrentdensityis
�=�
�??????=
��
�
??????
�
�=??????�
or �=??????�…………(5)ItisthepointformofOhm’slaw.
Where??????=
��
�
??????
�
istheconductivityoftheconductor.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Continuity Equation
•Thecontinuityequationisderivedfromlawconservationofcharge.Fromthe
lawofconservationcharge,thetimerateofdecreaseofchargewithinagiven
volumemustbeequaltothenetoutwardcurrentflowthroughtheclosed
surfaceofthevolume.Thatis
�
??????��=
−�??????
??????�
��
………….(1)
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Continuity Equation
•Thecontinuityequationisderivedfromlawconservationofcharge.Fromthe
lawofconservationcharge,thetimerateofdecreaseofchargewithinagiven
volumemustbeequaltothenetoutwardcurrentflowthroughtheclosed
surfaceofthevolume.Thatis
�
??????��=
−�??????
??????�
��
………….(1)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Continuity Equation cont.,
Aditya Engineering College (A)
Continuity Equation cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Capacitance
•ThecapacitanceCisthephysicalpropertyofthecapacitoranditismeasuredin
farads(F).
•ThecapacitanceCofthecapacitorisdefinedastheratioofthemagnitudeof
thechargeononeoftheplatestothepotentialdifferencebetweenthem.That
is
C=
??????
??????
………….(1)
•TypesofCapacitors:
A.Parallel–PlateCapacitors
B.CoaxialCapacitors
C.SphericalCapacitors
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Capacitance
•ThecapacitanceCisthephysicalpropertyofthecapacitoranditismeasuredin
farads(F).
•ThecapacitanceCofthecapacitorisdefinedastheratioofthemagnitudeof
thechargeononeoftheplatestothepotentialdifferencebetweenthem.That
is
C=
??????
??????
………….(1)
•TypesofCapacitors:
A.Parallel–PlateCapacitors
B.CoaxialCapacitors
C.SphericalCapacitors
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
A.Parallel –Plate Capacitors
•Consideraparallelplatecapacitorasshownbelow,thateachoftheplateshas
anareaSandtheyseparatedbyadistanced.
•ThecapacitanceCα
??????
�
=>�=
??????�
�
Where??????isthepermittivityofdielectricmaterial
Thatis??????=??????
0??????
??????
where??????
0isthepermittivityofthefreespace
??????
??????isthepermittivityofthefreespace
And??????
??????=
�
�0
,where�
0isthecapacitanceofcapacitorfilledwithair.
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A.Parallel –Plate Capacitors
•Consideraparallelplatecapacitorasshownbelow,thateachoftheplateshas
anareaSandtheyseparatedbyadistanced.
•ThecapacitanceCα
??????
�
=>�=
??????�
�
Where??????isthepermittivityofdielectricmaterial
Thatis??????=??????
0??????
??????
where??????
0isthepermittivityofthefreespace
??????
??????isthepermittivityofthefreespace
And??????
??????=
�
�0
,where�
0isthecapacitanceofcapacitorfilledwithair.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
B.Coaxial Capacitors
•Acoaxialcapacitorisessentiallyacoaxialcableorcoaxialcylindricalcapacitor.
•ConsiderlengthLoftwocoaxialconductorsofinnerradiusa�andouterradius
�(�>�)asshowninfigure.
•Thecapacitanceofacoaxialcapacitorisgivenby
�=
2�????????????
ln(
�
�
)
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B.Coaxial Capacitors
•Acoaxialcapacitorisessentiallyacoaxialcableorcoaxialcylindricalcapacitor.
•ConsiderlengthLoftwocoaxialconductorsofinnerradiusa�andouterradius
�(�>�)asshowninfigure.
•Thecapacitanceofacoaxialcapacitorisgivenby
�=
2�????????????
ln(
�
�
)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
C.Spherical Capacitors
•Asphericalcapacitoristhecaseoftwoconcentricconductors.
•Considerinnersphereofradius�andoutersphereradius�(�>�)separated
byadielectricmediumwithpermittivity??????asshowninfigure.asshownin
figure.
•Thecapacitanceofasphericalcapacitorisgivenby
�=
4�??????
1
�
−
1
�
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C.Spherical Capacitors
•Asphericalcapacitoristhecaseoftwoconcentricconductors.
•Considerinnersphereofradius�andoutersphereradius�(�>�)separated
byadielectricmediumwithpermittivity??????asshowninfigure.asshownin
figure.
•Thecapacitanceofasphericalcapacitorisgivenby
�=
4�??????
1
�
−
1
�
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
MAGNETOSTATICS
•Biot-Sarvat'sLaw
•Ampere’sLawMaxwell’sEquation
•MagneticfluxDensity-Maxwell’sEquation
•MagneticScalarandVectorPotentials
•ForcesduetoMagneticFields
•Inductance
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MAGNETOSTATICS
•Biot-Sarvat'sLaw
•Ampere’sLawMaxwell’sEquation
•MagneticfluxDensity-Maxwell’sEquation
•MagneticScalarandVectorPotentials
•ForcesduetoMagneticFields
•Inductance
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
MAGNETOSTATICS
Aditya Engineering College (A)
MAGNETOSTATICS
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
MAGNETOSTATICS
Aditya Engineering College (A)
MAGNETOSTATICS
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
MAGNETOSTATICS cont.,
Aditya Engineering College (A)
MAGNETOSTATICS cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Biot-Savart’sLaw
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Biot-Savart’sLaw
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Biot-Savart’sLaw cont.,
Aditya Engineering College (A)
Biot-Savart’sLaw cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Biot-Sarvat's Law cont.,
Aditya Engineering College (A)
Biot-Sarvat's Law cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Biot-Savart’sLaw cont.,
Aditya Engineering College (A)
Biot-Savart’sLaw cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Biot-Sarvat's Law cont.,
Aditya Engineering College (A)
Biot-Sarvat's Law cont.,
Tuesday, May 16, 2023EM Waves and Transmission Lines, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Biot-Sarvat's Law cont.,
Aditya Engineering College (A)
Biot-Sarvat's Law cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor
•ConsiderastraightcurrentcarryingfilamentaryconductoroffinitelengthAB,
along�-axiswithitsupperandlowerends,subtendingangles�
�and�
�at
pointP,thepointatwhichത�istobedetermined.
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Magnetic field ഥ�due to Straight filament conductor
•ConsiderastraightcurrentcarryingfilamentaryconductoroffinitelengthAB,
along�-axiswithitsupperandlowerends,subtendingangles�
�and�
�at
pointP,thepointatwhichത�istobedetermined.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor cont.,
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor cont.,
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor cont.,
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor cont.,
Aditya Engineering College (A)
Magnetic field ഥ�due to Straight filament conductor cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-1
Problem:Findഥ�at(0,0,5)duetoastraightcurrentcarryingconductoralongX-axis
oflength2mofcurrent10A.
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Example-1
Problem:Findഥ�at(0,0,5)duetoastraightcurrentcarryingconductoralongX-axis
oflength2mofcurrent10A.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-2
Derivetheexpressionformagneticfieldintensityത�atpointP(0,0,h)duetoa
circularringofcurrentelement�
2
+�
2
=�
2
,�=0carriesadirectcurrentofI.
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Example-2
Derivetheexpressionformagneticfieldintensityത�atpointP(0,0,h)duetoa
circularringofcurrentelement�
2
+�
2
=�
2
,�=0carriesadirectcurrentofI.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-2 cot.,
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Example-2 cot.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example -2 cont.,
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Example -2 cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-3
Acircularlooplocatedon�
2
+�
2
=9,�=0carriesadirectcurrentof10Aalong
�
∅
.Determineത�at(0,0,4)and(0,0,-4).
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Example-3
Acircularlooplocatedon�
2
+�
2
=9,�=0carriesadirectcurrentof10Aalong
�
∅
.Determineത�at(0,0,4)and(0,0,-4).
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Example-3 cont.,
Acircularlooplocatedon�
2
+�
2
=9,�=0carriesadirectcurrentof10Aalong
�
∅
.Determineത�at(0,0,4)and(0,0,-4).
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Example-3 cont.,
Acircularlooplocatedon�
2
+�
2
=9,�=0carriesadirectcurrentof10Aalong
�
∅
.Determineത�at(0,0,4)and(0,0,-4).
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Ampere’s Circuit law-Maxwell’s Equation
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Ampere’s Circuit law-Maxwell’s Equation
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Ampere’s Circuit law-Maxwell’s Equation cont.,
Aditya Engineering College (A)
Ampere’s Circuit law-Maxwell’s Equation cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic Flux Density-Maxwell’s Equation
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Magnetic Flux Density-Maxwell’s Equation
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic Flux Density-Maxwell’s Equation cont.,
Aditya Engineering College (A)
Magnetic Flux Density-Maxwell’s Equation cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Maxwell’s Equation-Static Electric and Magnetic Fields
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Maxwell’s Equation-Static Electric and Magnetic Fields
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic Scalar and Vector Potentials
Scalar Potential(�
??????)-Measured in Amperes
Magnetic Potential
Vector Potential(ҧ�)-Measured in Wb/m
Scalar Potential(�
�):
•Recall the vector identity, for a scalar quantity �
??????, we have
??????×−??????�
??????=0………(1)
•From Ampere’s law, we have ??????×�=�and for �=�, it becomes
??????×�=�………..(2)
From equation (1) and (2), we can write
�=−??????�
??????...........(3) ??????���:�����??????�����=−??????�
Where �
??????is called Magnetic Scalar Potential and it is existing for �=�.
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Magnetic Scalar and Vector Potentials
Scalar Potential(�
??????)-Measured in Amperes
Magnetic Potential
Vector Potential(ҧ�)-Measured in Wb/m
Scalar Potential(�
�):
•Recall the vector identity, for a scalar quantity �
??????, we have
??????×−??????�
??????=0………(1)
•From Ampere’s law, we have ??????×�=�and for �=�, it becomes
??????×�=�………..(2)
From equation (1) and (2), we can write
�=−??????�
??????...........(3) ??????���:�����??????�����=−??????�
Where �
??????is called Magnetic Scalar Potential and it is existing for �=�.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Magnetic Scalar and Vector Potentials cont.,
Vector Potential(ഥ�):
•Recall the vector identity, for a vector quantityഥ�, we have
??????.??????×ഥ�=0………(1)
•From the basics of magnetic fields, we have ??????.�=�………..(2)
From equation (1) and (2), we can write �=??????×ഥ�...........(3)
Where ഥ�is called Magnetic Vector Potential and it can be expressed as
ഥ�=
??????
??????�??????????????????
�????????????
………….(4)
But??????=
�
ഥ�.��=
�
(??????×ഥ�).��=ׯ
�
ഥ�.��(From Stoke’s theorem)
??????=ׯ
�
ഥ�.��………..(5)
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Magnetic Scalar and Vector Potentials cont.,
Vector Potential(ഥ�):
•Recall the vector identity, for a vector quantityഥ�, we have
??????.??????×ഥ�=0………(1)
•From the basics of magnetic fields, we have ??????.�=�………..(2)
From equation (1) and (2), we can write �=??????×ഥ�...........(3)
Where ഥ�is called Magnetic Vector Potential and it can be expressed as
ഥ�=
??????
??????�??????????????????
�????????????
………….(4)
But??????=
�
ഥ�.��=
�
(??????×ഥ�).��=ׯ
�
ഥ�.��(From Stoke’s theorem)
??????=ׯ
�
ഥ�.��………..(5)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Forces due to Magnetic Fields
There are three ways to experience the force due to magnetic fields.
1.Force on a moving charge in a magnetic field ത�
2.Force on a current element in a magnetic field ത�
3.Force between two current elements
1.Force on a moving charge in a magnetic field ഥ�:
•The electric force �
�on moving charge or stationary charge �in an electric field
ത�is give by �
�=�ത�……….(1)
•A magnetic field can exert force only on a moving charge.
•The magnetic force �
??????on moving charge �with velocity ത�is given by
�
??????=�ത�×ത�………(2)
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Forces due to Magnetic Fields
There are three ways to experience the force due to magnetic fields.
1.Force on a moving charge in a magnetic field ത�
2.Force on a current element in a magnetic field ത�
3.Force between two current elements
1.Force on a moving charge in a magnetic field ഥ�:
•The electric force �
�on moving charge or stationary charge �in an electric field
ത�is give by �
�=�ത�……….(1)
•A magnetic field can exert force only on a moving charge.
•The magnetic force �
??????on moving charge �with velocity ത�is given by
�
??????=�ത�×ത�………(2)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Forces due to Magnetic Fields cont.,
•The total force acting on a charge particle �in the presence of both electric
field ത�and magnetic field ത�is given by
�=�
�+�
??????
That is �=�ത�+�ത�×ത�………(3)
This is called Lorenz’s force equation.
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Forces due to Magnetic Fields cont.,
•The total force acting on a charge particle �in the presence of both electric
field ത�and magnetic field ത�is given by
�=�
�+�
??????
That is �=�ത�+�ത�×ത�………(3)
This is called Lorenz’s force equation.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Forces due to Magnetic Field cont.,
2.Force on a current element in a magnetic field ഥ�:
•The force acting on a current element ഥ��carrying current I in the presence of
magnetic field ത�is given by
��=Iഥ��×ത�
•The force acting on a current element of closed length L carrying current I in the
presence of magnetic field ത�is given by
That is ҧ�=ׯ
??????
Iഥ��×ത�………(3)
From the equation (3), we can define magnetic flux density ത�as
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Forces due to Magnetic Field cont.,
2.Force on a current element in a magnetic field ഥ�:
•The force acting on a current element ഥ��carrying current I in the presence of
magnetic field ത�is given by
��=Iഥ��×ത�
•The force acting on a current element of closed length L carrying current I in the
presence of magnetic field ത�is given by
That is ҧ�=ׯ
??????
Iഥ��×ത�………(3)
From the equation (3), we can define magnetic flux density ത�as
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Forces due to Magnetic Field cont.,
3.Force between two current elements:
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Forces due to Magnetic Field cont.,
3.Force between two current elements:
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Forces due to Magnetic Field cont.,
Aditya Engineering College (A)
Forces due to Magnetic Field cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Forces due to Magnetic Field cont.,
Aditya Engineering College (A)
Forces due to Magnetic Field cont.,
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Inductance and Inductor cont.,
•A closed circuit carrying current �produces a magnetic field ത�that causes a flux
??????=ത�.��to pass through each turn of the coil as shown in figure.
•If the circuit has N number of turns, the flux linkage λas
λ=N??????………..(1)
•The flux linkage λis proportional to current �as
λ∝�or λ=LI……..(2)
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Inductance and Inductor cont.,
•A closed circuit carrying current �produces a magnetic field ത�that causes a flux
??????=ത�.��to pass through each turn of the coil as shown in figure.
•If the circuit has N number of turns, the flux linkage λas
λ=N??????………..(1)
•The flux linkage λis proportional to current �as
λ∝�or λ=LI……..(2)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Inductance and Inductor cont.,
•A closed circuit carrying current �produces a magnetic field ത�that causes a flux
??????=ത�.��to pass through each turn of the coil as shown in figure.
If the circuit has N number of turns, the flux
linkage λas λ=N??????………..(1)
The flux linkage λis proportional to current
�as λ∝�or λ=LI……..(2)
•Where ??????is a constant of proportionality called the Self-Inductance of the
circuit. A circuit or part of a circuit that has inductance is called a inductor.
•The inductance of an inductor as the ratio of the magnetic flux linkage λto the
current �through the inductor., as??????=
λ
??????
=
????????????
??????
……….(3)
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Inductance and Inductor cont.,
•A closed circuit carrying current �produces a magnetic field ത�that causes a flux
??????=ത�.��to pass through each turn of the coil as shown in figure.
If the circuit has N number of turns, the flux
linkage λas λ=N??????………..(1)
The flux linkage λis proportional to current
�as λ∝�or λ=LI……..(2)
•Where ??????is a constant of proportionality called the Self-Inductance of the
circuit. A circuit or part of a circuit that has inductance is called a inductor.
•The inductance of an inductor as the ratio of the magnetic flux linkage λto the
current �through the inductor., as??????=
λ
??????
=
????????????
??????
……….(3)
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Inductance and Inductor cont.,
•The energy(magnetic) stored in an inductor is�
??????=
1
2
??????�
2
Mutual Inductance:
•If instead of having a single circuit, we have two circuits carrying current �
1and
�
2as shown in figure. Four fluxes ??????
11, ??????
12, ??????
21and ??????
22are produced.
•??????
12-the flux passing through circuit 1 due to current �
2in circuit 2.
•??????
21-the flux passing through circuit 2 due to current �
1in circuit 1.
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Inductance and Inductor cont.,
•The energy(magnetic) stored in an inductor is�
??????=
1
2
??????�
2
Mutual Inductance:
•If instead of having a single circuit, we have two circuits carrying current �
1and
�
2as shown in figure. Four fluxes ??????
11, ??????
12, ??????
21and ??????
22are produced.
•??????
12-the flux passing through circuit 1 due to current �
2in circuit 2.
•??????
21-the flux passing through circuit 2 due to current �
1in circuit 1.
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Inductance and Inductor cont.,
Mutual inductance (�
��):It is defined as the flux linkages of circuit 1 per unit
current �
2; that is
Mutual inductance (�
��):It is defined as the flux linkages of circuit 2 per unit
current �
1; that is
Aditya Engineering College (A)
Inductance and Inductor cont.,
Mutual inductance (�
��):It is defined as the flux linkages of circuit 1 per unit
current �
2; that is
Mutual inductance (�
��):It is defined as the flux linkages of circuit 2 per unit
current �
1; that is
Tuesday, May 16, 2023EM Waves and Transmission LinesL, UNIT-II Mr.T.SrinivasaRao, Associate Professor, ECE
Aditya Engineering College (A)
Energy stored in Capacitor and Inductor
•The electric energy stored in a capacitor �
�=
�
�
��
�
This also expressed as �
�=
�
�
�.���=
�
�
??????�
�
��
•The magnetic energy stored in an inductor �
�=
�
�
��
�
This also expressed as �
�=
�
�
�.���=
�
�
??????�
�
��
Aditya Engineering College (A)
Energy stored in Capacitor and Inductor
•The electric energy stored in a capacitor �
�=
�
�
��
�
This also expressed as �
�=
�
�
�.���=
�
�
??????�
�
��
•The magnetic energy stored in an inductor �
�=
�
�
��
�
This also expressed as �
�=
�
�
�.���=
�
�
??????�
�
��
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