1) Electromagnetic theory deals with the study of charges at rest and in motion and is fundamental to electrical engineering and physics. It is used in applications like RF communication, microwave engineering, antennas, electrical machines, and more.
2) Electromagnetic theory can be thought of as a generalization of circuit theory and is useful for situations that cannot be handled by circuit theory alone. It involves electric and magnetic field vectors rather than just voltages and currents.
3) Vector analysis is a useful mathematical tool for electromagnetic concepts. Important vector operations like addition, subtraction, scaling, dot product, and cross product are introduced. Common coordinate systems like Cartesian, cylindrical, and spherical polar coordinates are also discussed.
This first lecture describes what EMT is. Its history of evolution. Main personalities how discovered theories relating to this theory. Applications of EMT . Scalars and vectors and there algebra. Coordinate systems. Field, Coulombs law and electric field intensity.volume charge distribution, electric flux density, gauss's law and divergence
The International Journal of Engineering and Sciencetheijes
This document presents a method for identifying weak nodes and branches in electric power systems using metric projections. Metric projections are applied to the Jacobian matrix from a state estimator to calculate distances between nodes. Results show the voltages at each node decrease as reactive power is increased at one node. Metric projections identify nodes 1, 3 and 4 as weak based on their distances from other nodes in the Jacobian matrix. This technique can help utilities improve reactive power support and transmission capacity by identifying stressed areas of the system.
The document discusses vectors and vector arithmetic. It defines scalars and vectors, and explains that vectors have both magnitude and direction while scalars only have magnitude. It then discusses how vectors can be represented geometrically using arrows and how any two-dimensional vector can be decomposed into two components. The bulk of the document focuses on vector arithmetic, describing how vectors can be added and multiplied through various methods like the parallelogram law of addition, triangle law of addition, and scalar multiplication. It provides examples of how these vector arithmetic operations can be used to model real-world phenomena like displacements caused by earthquakes.
This document section provides an introduction to magnetic fields, including:
- Defining a magnetic field B as the force per unit charge on a moving charged particle.
- Explaining that magnetic force BF is perpendicular to both the particle's velocity v and the magnetic field B.
- Deriving the magnetic force on a current-carrying wire segment and using this to calculate the total magnetic force on wires of different shapes, including closed loops.
- Working through an example problem to calculate the magnetic forces on different parts of a semi-circular current loop placed in a uniform magnetic field.
This document provides an introduction to three-phase circuits and power. It defines key concepts like real power, reactive power, and power factor for sinusoidal voltages and currents. It describes how to calculate real and reactive power from rms voltage, current, and phase angle. Balanced three-phase systems are introduced, and how they allow more efficient power transmission compared to single-phase systems. Equations for solving problems involving three-phase circuits are also presented.
This document provides an introduction to tensor calculus for students with a background in linear algebra. It explains the key concepts of tensor calculus, including index notation, covariant and contravariant vectors, tensors, and the metric tensor. It focuses on practical applications of tensors in physics, especially in special and general relativity. The document covers definitions, properties, and calculus of tensors through examples and exercises.
This document discusses several topics related to steady electric currents:
- It defines different types of electric currents including conduction, electrolytic, convection, and displacement currents.
- It introduces current density and derives Ohm's law, relating current density to electric field and conductivity.
- It describes electromotive force (EMF) as the driving force provided by sources like batteries to maintain steady current in closed circuits, as stated by Kirchhoff's voltage law.
- It introduces the continuity equation relating divergence of current density to charge density changes, and derives Kirchhoff's current law equating the sum of currents at junctions to zero.
Application of differential equation in real lifeTanjil Hasan
Differential equations are used in many areas of real life including creating software, games, artificial intelligence, modeling natural phenomena, and providing theoretical explanations. Some examples given are using differential equations to model character velocity in games, understand computer hardware, solve constraint logic programs, describe physical laws, and model chemical reaction rates. Differential equations are an essential mathematical tool for describing how our world works.
This first lecture describes what EMT is. Its history of evolution. Main personalities how discovered theories relating to this theory. Applications of EMT . Scalars and vectors and there algebra. Coordinate systems. Field, Coulombs law and electric field intensity.volume charge distribution, electric flux density, gauss's law and divergence
The International Journal of Engineering and Sciencetheijes
This document presents a method for identifying weak nodes and branches in electric power systems using metric projections. Metric projections are applied to the Jacobian matrix from a state estimator to calculate distances between nodes. Results show the voltages at each node decrease as reactive power is increased at one node. Metric projections identify nodes 1, 3 and 4 as weak based on their distances from other nodes in the Jacobian matrix. This technique can help utilities improve reactive power support and transmission capacity by identifying stressed areas of the system.
The document discusses vectors and vector arithmetic. It defines scalars and vectors, and explains that vectors have both magnitude and direction while scalars only have magnitude. It then discusses how vectors can be represented geometrically using arrows and how any two-dimensional vector can be decomposed into two components. The bulk of the document focuses on vector arithmetic, describing how vectors can be added and multiplied through various methods like the parallelogram law of addition, triangle law of addition, and scalar multiplication. It provides examples of how these vector arithmetic operations can be used to model real-world phenomena like displacements caused by earthquakes.
This document section provides an introduction to magnetic fields, including:
- Defining a magnetic field B as the force per unit charge on a moving charged particle.
- Explaining that magnetic force BF is perpendicular to both the particle's velocity v and the magnetic field B.
- Deriving the magnetic force on a current-carrying wire segment and using this to calculate the total magnetic force on wires of different shapes, including closed loops.
- Working through an example problem to calculate the magnetic forces on different parts of a semi-circular current loop placed in a uniform magnetic field.
This document provides an introduction to three-phase circuits and power. It defines key concepts like real power, reactive power, and power factor for sinusoidal voltages and currents. It describes how to calculate real and reactive power from rms voltage, current, and phase angle. Balanced three-phase systems are introduced, and how they allow more efficient power transmission compared to single-phase systems. Equations for solving problems involving three-phase circuits are also presented.
This document provides an introduction to tensor calculus for students with a background in linear algebra. It explains the key concepts of tensor calculus, including index notation, covariant and contravariant vectors, tensors, and the metric tensor. It focuses on practical applications of tensors in physics, especially in special and general relativity. The document covers definitions, properties, and calculus of tensors through examples and exercises.
This document discusses several topics related to steady electric currents:
- It defines different types of electric currents including conduction, electrolytic, convection, and displacement currents.
- It introduces current density and derives Ohm's law, relating current density to electric field and conductivity.
- It describes electromotive force (EMF) as the driving force provided by sources like batteries to maintain steady current in closed circuits, as stated by Kirchhoff's voltage law.
- It introduces the continuity equation relating divergence of current density to charge density changes, and derives Kirchhoff's current law equating the sum of currents at junctions to zero.
Application of differential equation in real lifeTanjil Hasan
Differential equations are used in many areas of real life including creating software, games, artificial intelligence, modeling natural phenomena, and providing theoretical explanations. Some examples given are using differential equations to model character velocity in games, understand computer hardware, solve constraint logic programs, describe physical laws, and model chemical reaction rates. Differential equations are an essential mathematical tool for describing how our world works.
3-1 VECTORS AND THEIR COMPONENTS
After reading this module, you should be able to . . .
3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws.
3.02 Subtract a vector from a second one.
3.03 Calculate the components of a vector on a given coordinate system, showing them in a drawing.
3.04 Given the components of a vector, draw the vector
and determine its magnitude and orientation.
3.05 Convert angle measures between degrees and radians.
3-2 UNIT VECTORS, ADDING VECTORS BY COMPONENTS
After reading this module, you should be able to . . .
3.06 Convert a vector between magnitude-angle and unit vector notations.
3.07 Add and subtract vectors in magnitude-angle notation
and in unit-vector notation.
3.08 Identify that, for a given vector, rotating the coordinate
system about the origin can change the vector’s components but not the vector itself.
etc...
This document provides an introduction to partial differential equations (PDEs). It defines PDEs as equations that contain derivatives of unknown functions of several variables and one or more partial derivatives. The solutions to PDEs are differentiable functions that satisfy boundary or initial conditions. PDEs are often used to express laws of physics. Examples of common PDEs discussed include the Laplace equation, Poisson equation, wave equation, heat equation, and diffusion equation.
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to …
2.01 Identify that if all parts of an object move in the same direction and at the same rate, we can treat the object as if it
were a (point-like) particle. (This chapter is about the motion of such objects.)
2.02 Identify that the position of a particle is its location as
read on a scaled axis, such as an x-axis.
2.03 Apply the relationship between a particle’s
displacement and its initial and final positions.
2.04 Apply the relationship between a particle’s average
velocity, its displacement, and the time interval for that
displacement.
2.05 Apply the relationship between a particle’s average
speed, the total distance it moves, and the time interval for
the motion.
2.06 Given a graph of a particle’s position versus time,
determine the average velocity between any two particular
times.
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to . . .
2.07 Given a particle’s position as a function of time,
calculate the instantaneous velocity for any particular time.
2.08 Given a graph of a particle’s position versus time, determine the instantaneous velocity for any particular time.
2.09 Identify speed as the magnitude of the instantaneous
velocity.
etc......
2-Dimensional and 3-Dimesional Electromagnetic Fields Using Finite element me...IOSR Journals
This document describes using the finite element method to model 2D and 3D electromagnetic fields. It discusses modeling a quarter section of a rectangular coaxial line with triangular elements. It describes constructing the matrices for each element and combining them to solve the overall matrix equation. The document outlines implementing FEM in MATLAB, including generating meshes, adding sources, and solving the resulting matrices. Several examples are presented of using a graphical user interface created in MATLAB to calculate fields from configurations like straight wires, bent wires, solenoids, and square loops using FEM techniques.
This document discusses a quarter-symmetric non-metric connection in a Lorentzian β-Kenmotsu manifold. It begins by introducing quarter-symmetric connections and defining the specific quarter-symmetric non-metric connection to be studied. Properties of this connection such as its torsion tensor and covariant derivative of 1-forms are derived. The curvature tensor and first Bianchi identity are obtained for the Riemannian manifold with respect to this quarter-symmetric non-metric connection. Finally, the Ricci tensor, scalar curvature, and some torsion tensor identities are calculated for the Lorentzian β-Kenmotsu manifold with respect to this connection.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
This document discusses key concepts related to centroid and moment of inertia including:
- Definitions of centroid, center of gravity, and center of mass
- Methods for determining the centroid of areas, lines, volumes, and composite bodies using integration
- The perpendicular axis theorem and parallel axis theorem for calculating moments of inertia
- Equations for calculating the polar moment of inertia and moments of inertia of simple and composite areas/bodies using integration
1) The document analyzes the scattering properties of a conducting cylinder coated with an anisotropic chiral material using Mie's approach. Boundary conditions are used to derive a set of equations relating the scattered field coefficients, which are solved numerically.
2) Differential scattering cross sections are calculated for co-polarized and cross-polarized waves. Numerical results show the influence of chirality, permittivity, and permeability on the scattering response. Increasing chiral coating thickness decreases the co-polarized cross section.
3) Comparisons with previous work on a perfectly conducting chiral coated cylinder show good agreement. The scattering pattern has a maximum in the forward direction, which decreases with increasing chirality.
1. Static electricity can be demonstrated by rubbing a glass rod with silk and hanging it from a thread. A second glass rod rubbed with silk will repel the first, while a hard rubber rod rubbed with fur will attract the glass rod.
2. The force between two electrically charged bodies follows an inverse square law, where the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
3. Electric potential is defined as the work required to move a unit positive charge from a reference point (usually infinity) to the point where the potential is being measured, without producing an acceleration.
6161103 9.2 center of gravity and center of mass and centroid for a bodyetcenterrbru
This document discusses the center of gravity, center of mass, and centroid of rigid bodies. It defines these terms and presents methods to calculate them using integrals of differential elements. Examples are provided to demonstrate calculating the centroid for areas and lines using appropriate coordinate systems and differential elements. Centroids are found by taking moments of these elements and the document outlines the general procedure to perform these calculations.
- Strain gauges oriented in different directions and arranged in a rosette configuration are used to determine principal strains and stresses at a point.
- Several types of rosette configurations are commonly used, including rectangular, delta, and tee rosettes with varying numbers and orientations of strain gauges.
- Simultaneous equations relating strain readings from each gauge to the principal strains are solved to determine the principal strain values and orientations.
- Principal stresses are then calculated from the principal strains using stress-strain relationships accounting for Poisson's ratio.
Numerical disperison analysis of sympletic and adi schemexingangahu
This document discusses numerical dispersion analysis of symplectic and alternating direction implicit (ADI) schemes for computational electromagnetic simulation. It presents Maxwell's equations as a Hamiltonian system that can be written as symplectic or ADI schemes by approximating the time evolution operator. Three high order spatial difference approximations - high order staggered difference, compact finite difference, and scaling function approximations - are analyzed to reduce numerical dispersion when combined with the symplectic and ADI schemes. The document derives unified dispersion relationships for the symplectic and ADI schemes with different spatial difference approximations, which can be used as a reference for simulating large scale electromagnetic problems.
Elextrostatic and Electromagnetic Fields Actuators for MEMS AD/DA ConvertersCSCJournals
MEMS Analog -to-digital and digital-to- analog converters are proposed using electrostatic field and electromagnetic fields actuators. For the former, parallel deformable plates supported by springs are used with bias applied voltage which determines the amount of static displacement needed for equilibrium condition. For the latter, coil winding(s) are embedded in a rotating plate, which is exposed to a constant field of a permanent magnet, causing the plate to deflect according to the currents in the windings. In the analog-to-digital arrangement, different spring displacements are tapped off either the spring in case of electrostatic or the moving plate in case of electromagnetic actuators, corresponding to the binary decoded currents. At these off tapping points, logic high signal levels are applied at these locations so that when a certain analog voltage is applied on the moving plate of the capacitor, the spring is displaced to one of these locations, enabling different binary voltages to all switches up to that level. Similar result occurs when an analog voltage is applied on the winding. The digital binary voltages are fed to a priority encoder to obtain the digital value. In digital-to-analog arrangement, the input binary voltage is decoded to different spring locations which correspond to resistances making up a potentiometer circuit for the output analog voltage. Similarly; for the electromagnetic actuator, a number of different length coil windings are embedded within the moving plate, causing different deflections corresponding to one bit of the binary input.
Electromagnetic fields: Review of vector algebraDr.SHANTHI K.G
This document provides an introduction to electromagnetic fields and vector algebra concepts. It begins with an overview of vector algebra topics like vector addition, multiplication of vectors by scalars, and dot and cross products. It then discusses orthogonal coordinate systems, focusing on Cartesian coordinates. The document provides examples and solved problems for various vector algebra concepts. It aims to review key vector algebra that will be used as a mathematical tool for electromagnetic concepts.
This paper deals with certain configuration spaces
where the underlying geometry is sub-Riemannian because of the
physical constraints arising out of the mechanical systems we are
interested in. A motivated introduction to Sub-Riemannian
structures is included following which we look at the broad
science of mechanics where in the sub-Riemannian geometric
study aids us to talk about applications like robotics and image
analysis.
The document discusses the concept of the center of gravity or centroid of a body. It defines the center of gravity as the point where the entire weight of a body can be considered to be concentrated. The center of gravity is determined by the distribution of mass in the body. The document outlines different methods for calculating the centroid based on whether the body can be modeled as a line, area, or volume. It also notes that if a body has an axis of symmetry, its centroid must lie along that axis, or at the intersection of axes if it has multiple symmetries.
This document describes electric potential and its relationship to electric field and potential energy. It begins by introducing electric potential and defining it as the work required per unit charge to move a test charge between two points against an electric field. The electric potential due to point charges and continuous charge distributions is then derived. Methods for calculating electric potential and field from each other are presented, along with examples such as charged rods, rings, and disks. The chapter concludes with problem-solving strategies and additional practice problems.
Mohawk College is accepting applications for its May 2014 intake. It offers diverse programs at the apprenticeship, continuing education, diploma and degree levels to optimize students' education. Cooperative education programs give students real-world experience. Mohawk College is accredited and funded by the Ministry of Training, Colleges and Universities of Canada. It lists admission requirements and over 50 programs, including degrees in nursing, medical radiation sciences, and advanced diplomas/diplomas in various fields like engineering, business, health, and technology. Contact information is provided for Dhrron Consultancy for more details.
The Saskatchewan Institute of Applied Science and Technology is Saskatchewan's primary public institution for post-secondary technical education and skills training. It offers a variety of diploma and certificate programs in fields like business, engineering, health sciences, and skilled trades. Admission requirements include a minimum 50% average from previous education and meeting English language proficiency standards. The document provides contact information for Dhrron Consultancy, an organization that can provide more details on admission and programs.
3-1 VECTORS AND THEIR COMPONENTS
After reading this module, you should be able to . . .
3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws.
3.02 Subtract a vector from a second one.
3.03 Calculate the components of a vector on a given coordinate system, showing them in a drawing.
3.04 Given the components of a vector, draw the vector
and determine its magnitude and orientation.
3.05 Convert angle measures between degrees and radians.
3-2 UNIT VECTORS, ADDING VECTORS BY COMPONENTS
After reading this module, you should be able to . . .
3.06 Convert a vector between magnitude-angle and unit vector notations.
3.07 Add and subtract vectors in magnitude-angle notation
and in unit-vector notation.
3.08 Identify that, for a given vector, rotating the coordinate
system about the origin can change the vector’s components but not the vector itself.
etc...
This document provides an introduction to partial differential equations (PDEs). It defines PDEs as equations that contain derivatives of unknown functions of several variables and one or more partial derivatives. The solutions to PDEs are differentiable functions that satisfy boundary or initial conditions. PDEs are often used to express laws of physics. Examples of common PDEs discussed include the Laplace equation, Poisson equation, wave equation, heat equation, and diffusion equation.
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to …
2.01 Identify that if all parts of an object move in the same direction and at the same rate, we can treat the object as if it
were a (point-like) particle. (This chapter is about the motion of such objects.)
2.02 Identify that the position of a particle is its location as
read on a scaled axis, such as an x-axis.
2.03 Apply the relationship between a particle’s
displacement and its initial and final positions.
2.04 Apply the relationship between a particle’s average
velocity, its displacement, and the time interval for that
displacement.
2.05 Apply the relationship between a particle’s average
speed, the total distance it moves, and the time interval for
the motion.
2.06 Given a graph of a particle’s position versus time,
determine the average velocity between any two particular
times.
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to . . .
2.07 Given a particle’s position as a function of time,
calculate the instantaneous velocity for any particular time.
2.08 Given a graph of a particle’s position versus time, determine the instantaneous velocity for any particular time.
2.09 Identify speed as the magnitude of the instantaneous
velocity.
etc......
2-Dimensional and 3-Dimesional Electromagnetic Fields Using Finite element me...IOSR Journals
This document describes using the finite element method to model 2D and 3D electromagnetic fields. It discusses modeling a quarter section of a rectangular coaxial line with triangular elements. It describes constructing the matrices for each element and combining them to solve the overall matrix equation. The document outlines implementing FEM in MATLAB, including generating meshes, adding sources, and solving the resulting matrices. Several examples are presented of using a graphical user interface created in MATLAB to calculate fields from configurations like straight wires, bent wires, solenoids, and square loops using FEM techniques.
This document discusses a quarter-symmetric non-metric connection in a Lorentzian β-Kenmotsu manifold. It begins by introducing quarter-symmetric connections and defining the specific quarter-symmetric non-metric connection to be studied. Properties of this connection such as its torsion tensor and covariant derivative of 1-forms are derived. The curvature tensor and first Bianchi identity are obtained for the Riemannian manifold with respect to this quarter-symmetric non-metric connection. Finally, the Ricci tensor, scalar curvature, and some torsion tensor identities are calculated for the Lorentzian β-Kenmotsu manifold with respect to this connection.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
This document discusses key concepts related to centroid and moment of inertia including:
- Definitions of centroid, center of gravity, and center of mass
- Methods for determining the centroid of areas, lines, volumes, and composite bodies using integration
- The perpendicular axis theorem and parallel axis theorem for calculating moments of inertia
- Equations for calculating the polar moment of inertia and moments of inertia of simple and composite areas/bodies using integration
1) The document analyzes the scattering properties of a conducting cylinder coated with an anisotropic chiral material using Mie's approach. Boundary conditions are used to derive a set of equations relating the scattered field coefficients, which are solved numerically.
2) Differential scattering cross sections are calculated for co-polarized and cross-polarized waves. Numerical results show the influence of chirality, permittivity, and permeability on the scattering response. Increasing chiral coating thickness decreases the co-polarized cross section.
3) Comparisons with previous work on a perfectly conducting chiral coated cylinder show good agreement. The scattering pattern has a maximum in the forward direction, which decreases with increasing chirality.
1. Static electricity can be demonstrated by rubbing a glass rod with silk and hanging it from a thread. A second glass rod rubbed with silk will repel the first, while a hard rubber rod rubbed with fur will attract the glass rod.
2. The force between two electrically charged bodies follows an inverse square law, where the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
3. Electric potential is defined as the work required to move a unit positive charge from a reference point (usually infinity) to the point where the potential is being measured, without producing an acceleration.
6161103 9.2 center of gravity and center of mass and centroid for a bodyetcenterrbru
This document discusses the center of gravity, center of mass, and centroid of rigid bodies. It defines these terms and presents methods to calculate them using integrals of differential elements. Examples are provided to demonstrate calculating the centroid for areas and lines using appropriate coordinate systems and differential elements. Centroids are found by taking moments of these elements and the document outlines the general procedure to perform these calculations.
- Strain gauges oriented in different directions and arranged in a rosette configuration are used to determine principal strains and stresses at a point.
- Several types of rosette configurations are commonly used, including rectangular, delta, and tee rosettes with varying numbers and orientations of strain gauges.
- Simultaneous equations relating strain readings from each gauge to the principal strains are solved to determine the principal strain values and orientations.
- Principal stresses are then calculated from the principal strains using stress-strain relationships accounting for Poisson's ratio.
Numerical disperison analysis of sympletic and adi schemexingangahu
This document discusses numerical dispersion analysis of symplectic and alternating direction implicit (ADI) schemes for computational electromagnetic simulation. It presents Maxwell's equations as a Hamiltonian system that can be written as symplectic or ADI schemes by approximating the time evolution operator. Three high order spatial difference approximations - high order staggered difference, compact finite difference, and scaling function approximations - are analyzed to reduce numerical dispersion when combined with the symplectic and ADI schemes. The document derives unified dispersion relationships for the symplectic and ADI schemes with different spatial difference approximations, which can be used as a reference for simulating large scale electromagnetic problems.
Elextrostatic and Electromagnetic Fields Actuators for MEMS AD/DA ConvertersCSCJournals
MEMS Analog -to-digital and digital-to- analog converters are proposed using electrostatic field and electromagnetic fields actuators. For the former, parallel deformable plates supported by springs are used with bias applied voltage which determines the amount of static displacement needed for equilibrium condition. For the latter, coil winding(s) are embedded in a rotating plate, which is exposed to a constant field of a permanent magnet, causing the plate to deflect according to the currents in the windings. In the analog-to-digital arrangement, different spring displacements are tapped off either the spring in case of electrostatic or the moving plate in case of electromagnetic actuators, corresponding to the binary decoded currents. At these off tapping points, logic high signal levels are applied at these locations so that when a certain analog voltage is applied on the moving plate of the capacitor, the spring is displaced to one of these locations, enabling different binary voltages to all switches up to that level. Similar result occurs when an analog voltage is applied on the winding. The digital binary voltages are fed to a priority encoder to obtain the digital value. In digital-to-analog arrangement, the input binary voltage is decoded to different spring locations which correspond to resistances making up a potentiometer circuit for the output analog voltage. Similarly; for the electromagnetic actuator, a number of different length coil windings are embedded within the moving plate, causing different deflections corresponding to one bit of the binary input.
Electromagnetic fields: Review of vector algebraDr.SHANTHI K.G
This document provides an introduction to electromagnetic fields and vector algebra concepts. It begins with an overview of vector algebra topics like vector addition, multiplication of vectors by scalars, and dot and cross products. It then discusses orthogonal coordinate systems, focusing on Cartesian coordinates. The document provides examples and solved problems for various vector algebra concepts. It aims to review key vector algebra that will be used as a mathematical tool for electromagnetic concepts.
This paper deals with certain configuration spaces
where the underlying geometry is sub-Riemannian because of the
physical constraints arising out of the mechanical systems we are
interested in. A motivated introduction to Sub-Riemannian
structures is included following which we look at the broad
science of mechanics where in the sub-Riemannian geometric
study aids us to talk about applications like robotics and image
analysis.
The document discusses the concept of the center of gravity or centroid of a body. It defines the center of gravity as the point where the entire weight of a body can be considered to be concentrated. The center of gravity is determined by the distribution of mass in the body. The document outlines different methods for calculating the centroid based on whether the body can be modeled as a line, area, or volume. It also notes that if a body has an axis of symmetry, its centroid must lie along that axis, or at the intersection of axes if it has multiple symmetries.
This document describes electric potential and its relationship to electric field and potential energy. It begins by introducing electric potential and defining it as the work required per unit charge to move a test charge between two points against an electric field. The electric potential due to point charges and continuous charge distributions is then derived. Methods for calculating electric potential and field from each other are presented, along with examples such as charged rods, rings, and disks. The chapter concludes with problem-solving strategies and additional practice problems.
Mohawk College is accepting applications for its May 2014 intake. It offers diverse programs at the apprenticeship, continuing education, diploma and degree levels to optimize students' education. Cooperative education programs give students real-world experience. Mohawk College is accredited and funded by the Ministry of Training, Colleges and Universities of Canada. It lists admission requirements and over 50 programs, including degrees in nursing, medical radiation sciences, and advanced diplomas/diplomas in various fields like engineering, business, health, and technology. Contact information is provided for Dhrron Consultancy for more details.
The Saskatchewan Institute of Applied Science and Technology is Saskatchewan's primary public institution for post-secondary technical education and skills training. It offers a variety of diploma and certificate programs in fields like business, engineering, health sciences, and skilled trades. Admission requirements include a minimum 50% average from previous education and meeting English language proficiency standards. The document provides contact information for Dhrron Consultancy, an organization that can provide more details on admission and programs.
Terra Contracting uses six principles - integrity, safety, customer focus, intensity, training, and teamwork - to deliver high-quality projects and grow its business. Founded by Steve Taplin, who learned business lessons from his father, Terra has grown from a small family business into a $40 million company. Taplin credits the six principles for Terra's success and emphasizes their importance in decision making. Terra was recently acquired by Great Lakes Dredge & Dock for $20 million, which will help Terra take on larger projects through access to more resources.
This document is an outline for a lecture on drug abuse given at Pannasastra University of Cambodia. It defines drug addiction and outlines the biological, psychological, and social factors that can lead to drug abuse. It discusses the symptoms of drug addiction and potential complications. It also covers prevention methods and treatment options. The conclusion states that drug abuse damages health, behavior, and relationships while also impacting society through crime and costs of treatment.
This chapter discusses static fluid properties including pressure, units of pressure, pressure measurement instruments, the manometric equation, calculation of pressure forces on submerged surfaces, buoyancy force calculation, and fluid equilibrium. It is important that all applications discussed assume the fluid is at rest. The document provides examples and exercises to illustrate these static fluid concepts.
A humorous look at some office behaviour!
This was my first time showing my ipad drawings during a business presentation... and it worked really well. Ok the humour helped but there was a serious message to get across about respect for others especially in open plan offices
This document summarizes the admission requirements and programs offered at Confederation College in Canada. Applicants must be 19 years or older and have an HSC, AISSCE, PU or intermediate certificate from India. English language requirements are an IELTS score of 6.5 overall and 5.5 in each module, or 6.0 for health programs. The college offers over 80 programs across various fields including aboriginal studies, aviation, business, community services, engineering, general arts and science, health sciences, hospitality, media arts, natural resources, protective services, skilled trades, and access and upgrading programs. Students who join Dhrron Consultancy receive benefits like free IELTS coaching, career counseling, accommodation
This document summarizes the key findings of the 2013 McGladrey Manufacturing & Distribution Monitor. It provides an overview of Scott Bjornstad from McGladrey LLP who presented the Monitor findings. The Monitor surveyed over 1,000 manufacturing and distribution businesses. Key findings included that business optimism increased from 2012 to 2013, regulatory policies were viewed as limiting growth, sales increased for most businesses, costs were expected to rise, and thriving businesses planned to increase hiring in the next year. Workforce challenges around finding skilled talent and increasing employee costs were also discussed.
This document discusses different types of love between people including allowance, help, friendship, generosity, joy, smiles, dreams, choice, unity, and cohesion. It touches on photography and expresses gratitude for attention.
Pengaruh Posisi Intruder terhadap Bentuk Permukaan Bed Granular pada Efek Kac...Sparisoma Viridi
Penelitian ini meneliti pengaruh posisi intruder terhadap bentuk permukaan bed granular pada efek kacang brasil dua dimensi. Hasilnya menunjukkan bahwa bentuk permukaan bed dipengaruhi oleh posisi intruder, dan jarak pusat kelengkungan permukaan ke intruder berhubungan kuadratik dengan posisi vertikal intruder. Kelengkungan permukaan awal dan akhir lebih rendah dibandingkan kelengkungan bagian tengah.
This document analyzes news consumption trends on the internet from 2006-2007. It finds that while some large, national brand name newspapers and television networks saw increased online traffic, many local newspapers and television stations saw stagnant or declining traffic. It also finds that non-traditional news outlets like search engines, aggregators, and blogs are growing faster and attract more overall traffic than traditional news organizations. The challenges for newspapers are compounded by the fact that replacing print readers with online readers is much less profitable. The conclusion speculates that these trends could impact the future of news in America as online consumption continues to rise.
Smarthphone Berbasis Android: Pembelajaran Fisika dengan EksperimenSparisoma Viridi
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Association CEOs who have introduced new and innovative affinity programs over the past year. The show us the money panel.
Bill Carteaux, President & CEO, SPI: The Plastics Industry Trade Association
Kraig Naasz, President and CEO, American Frozen Food Institute
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- The document discusses static electric fields and key concepts like Coulomb's Law, Gauss's Law, electric potential, and boundary conditions.
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Some Research Notes on developing a Hybrid UAV for space industrialization. Goal is to develop profitable routes, infrastructure and vehicles to harvest power from Venus, Mercury and Sun and transmit power to interests
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Fundamental Concepts on Electromagnetic TheoryAL- AMIN
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In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Best 20 SEO Techniques To Improve Website Visibility In SERPPixlogix Infotech
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GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
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HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
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Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
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As the digital landscape continually evolves, operating systems play a critical role in shaping user experiences and productivity. The launch of Nitrux Linux 3.5.0 marks a significant milestone, offering a robust alternative to traditional systems such as Windows 11. This article delves into the essence of Nitrux Linux 3.5.0, exploring its unique features, advantages, and how it stands as a compelling choice for both casual users and tech enthusiasts.
TrustArc Webinar - 2024 Global Privacy SurveyTrustArc
How does your privacy program stack up against your peers? What challenges are privacy teams tackling and prioritizing in 2024?
In the fifth annual Global Privacy Benchmarks Survey, we asked over 1,800 global privacy professionals and business executives to share their perspectives on the current state of privacy inside and outside of their organizations. This year’s report focused on emerging areas of importance for privacy and compliance professionals, including considerations and implications of Artificial Intelligence (AI) technologies, building brand trust, and different approaches for achieving higher privacy competence scores.
See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
This webinar will review:
- The top 10 privacy insights from the fifth annual Global Privacy Benchmarks Survey
- The top challenges for privacy leaders, practitioners, and organizations in 2024
- Key themes to consider in developing and maintaining your privacy program
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
1. Mathematical Fundamentals
Introduction :
Electromagnetic theory is a discipline concerned with the study of charges at rest and in motion. Electromagnetic
principles are fundamental to the study of electrical engineering and physics. Electromagnetic theory is also
indispensable to the understanding, analysis and design of various electrical, electromechanical and electronic
systems. Some of the branches of study where electromagnetic principles find application are:
RF communication
Microwave Engineering
Antennas
Electrical Machines
Satellite Communication
Atomic and nuclear research
Radar Technology
Remote sensing
EMI EMC
Quantum Electronics
VLSI
Electromagnetic theory is a prerequisite for a wide spectrum of studies in the field of Electrical
Sciences and Physics. Electromagnetic theory can be thought of as generalization of circuit
theory. There are certain situations that can be handled exclusively in terms of field theory. In
electromagnetic theory, the quantities involved can be categorized assource quantities and field
quantities. Source of electromagnetic field is electric charges: either at rest or in motion.
However an electromagnetic field may cause a redistribution of charges that in turn change the
field and hence the separation of cause and effect is not always visible.
Electric charge is a fundamental property of matter. Charge exist only in positive or negative
integral multiple ofelectronic charge, -e, e= 1.60 × 10-19 coulombs. [It may be noted here that in
1962, Murray Gell-Mann hypothesized Quarks as the basic building blocks of matters. Quarks
were predicted to carry a fraction of electronic charge and the existence of Quarks have been
experimentally verified.] Principle of conservation of charge states that the total charge (algebraic
sum of positive and negative charges) of an isolated system remains unchanged, though the
charges may redistribute under the influence of electric field. Kirchhoff's Current Law (KCL) is an
assertion of the conservative property of charges under the implicit assumption that there is no
accumulation of charge at the junction.
2. Electromagnetic theory deals directly with the electric and magnetic field vectors where as circuit
theory deals with the voltages and currents. Voltages and currents are integrated effects of
electric and magnetic fields respectively. Electromagnetic field problems involve three space
variables along with the time variable and hence the solution tends to become correspondingly
complex. Vector analysis is a mathematical tool with which electromagnetic concepts are more
conveniently expressed and best comprehended. Since use of vector analysis in the study of
electromagnetic field theory results in real economy of time and thought, we first introduce the
concept of vector analysis.
Vector Analysis:
The quantities that we deal in electromagnetic theory may be either scalar or vectors [There are
other class of physical quantities called Tensors: where magnitude and direction vary with co
ordinate axes]. Scalars are quantities characterized by magnitude only and algebraic sign. A
quantity that has direction as well as magnitude is called a vector. Both scalar and vector
quantities are function of time and position . A field is a function that specifies a particular quantity
everywhere in a region. Depending upon the nature of the quantity under consideration, the field
may be a vector or a scalar field. Example of scalar field is the electric potential in a region while
electric or magnetic fields at any point is the example of vector field.
A vector can be written as, , where, is the magnitude and is the unit
vector which has unit magnitude and same direction as that of .
Two vector and are added together to give another vector . We have
................(1.1)
Let us see the animations in the next pages for the addition of two vectors, which has two rules:
1: Parallelogram law and 2: Head & tail rule
Vector Subtraction is similarly carried out: ........................(1.2)
3. Scaling of a vector is defined as , where is scaled version of vector and is a
scalar.
Some important laws of vector algebra are:
Commutative Law..........................................(1.3)
Associative Law.............................................(1.4)
Distributive Law ............................................(1.5)
The position vector of a point P is the directed distance from the origin (O) to P, i.e., =
.
Fig 1.3: Distance Vector
If = OP and = OQ are the position vectors of the points P and Q then the distance vector
4. Product of Vectors
When two vectors and are multiplied, the result is either a scalar or a vector depending
how the two vectors were multiplied. The two types of vector multiplication are:
Scalar product (or dot product) gives a scalar.
Vector product (or cross product) gives a vector.
The dot product between two vectors is defined as = |A||B|cosθAB ..................(1.6)
Vector product
is unit vector perpendicular to and
Fig 1.4 : Vector dot product
The dot product is commutative i.e., and distributive i.e., .
Associative law does not apply to scalar product.
The vector or cross product of two vectors and is denoted by . is a vector
perpendicular to the plane containing and , the magnitude is given by and
direction is given by right hand rule as explained in Figure 1.5.
............................................................................................(1.7)
where is the unit vector given by, .
The following relations hold for vector product.
5. = i.e., cross product is non commutative ..........(1.8)
i.e., cross product is distributive.......................(1.9)
i.e., cross product is non associative..............(1.10)
Scalar and vector triple product :
Scalar triple product .................................(1.11)
Vector triple product ...................................(1.12)
Co-ordinate Systems
In order to describe the spatial variations of the quantities, we require using appropriate co-
ordinate system. A point or vector can be represented in a curvilinear coordinate system that
may be orthogonal or non-orthogonal .
An orthogonal system is one in which the co-ordinates are mutually perpendicular. Non-
orthogonal co-ordinate systems are also possible, but their usage is very limited in practice .
Let u = constant, v = constant and w = constant represent surfaces in a coordinate system, the
surfaces may be curved surfaces in general. Furthur, let , and be the unit vectors in the
three coordinate directions(base vectors). In a general right handed orthogonal curvilinear
systems, the vectors satisfy the following relations :
.....................................(1.13)
These equations are not independent and specification of one will automatically imply the other
two. Furthermore, the following relations hold
................(1.14)
6. A vector can be represented as sum of its orthogonal components,
...................(1.15)
In general u, v and w may not represent length. We multiply u, v and w by conversion
factors h1,h2 and h3 respectively to convert differential changes du, dv and dw to corresponding
changes in length dl1, dl2, and dl3. Therefore
...............(1.16)
In the same manner, differential volume dv can be written as and differential
area ds1 normal to is given by, . In the same manner, differential areas normal
to unit vectors and can be defined.
In the following sections we discuss three most commonly used orthogonal co-ordinate
systems, viz:
1. Cartesian (or rectangular) co-ordinate system
2. Cylindrical co-ordinate system
3. Spherical polar co-ordinate system
Cartesian Co-ordinate System :
In Cartesian co-ordinate system, we have, (u,v,w) = (x,y,z). A point P(x0, y0, z0) in Cartesian co-
ordinate system is represented as intersection of three planes x = x0, y = y0 and z = z0. The unit
vectors satisfies the following relation:
7. Fig 1.6:Cartesian co-ordinate system
....................(1.17)
.....................(1.18)
In cartesian co-ordinate system, a vector can be written as . The dot
and cross product of two vectors and can be written as follows:
.................(1.19)
8. ....................(1.20)
Since x, y and z all represent lengths, h1= h2= h3=1. The differential length, area and volume are
defined respectively as
................(1.21)
.................................(1.22)
Cylindrical Co-ordinate System :
For cylindrical coordinate systems we have a point is determined
as the point of intersection of a cylindrical surface r = r0, half plane containing the z-axis and
making an angle ; with the xz plane and a plane parallel to xy plane located at z=z0 as
shown in figure 7 on next page.
In cylindrical coordinate system, the unit vectors satisfy the following relations
.....................(1.23)
A vector can be written as , ...........................(1.24)
9. The differential length is defined as,
......................(1.25)
Fig 1.7 : Cylindrical Coordinate System
11. Transformation between Cartesian and Cylindrical coordinates:
Let us consider is to be expressed in Cartesian co-ordinate
as . In doing so we note that
and it applies for other components as well.
Fig 1.9 : Unit Vectors in Cartesian and Cylindrical Coordinates
...............(1.28)
Therefore we can write, ..........(1.29)
12. These relations can be put conveniently in the matrix form as:
.....................(1.30)
themselves may be functions of as:
............................(1.31)
The inverse relationships are: ........................(1.32)
Fig 1.10: Spherical Polar Coordinate System
13. Thus we see that a vector in one coordinate system is transformed to another coordinate system
through two-step process: Finding the component vectors and then variable transformation.
Spherical Polar Coordinates:
For spherical polar coordinate system, we have, . A point is
represented as the intersection of
(i) Spherical surface r=r0
(ii) Conical surface ,and
(iii) half plane containing z-axis making angle with the xz plane as shown in the figure 1.10.
The unit vectors satisfy the following relationships: .....................................(1.33)
The orientation of the unit vectors are shown in the figure 1.11.
14. Fig 1.11: Orientation of Unit Vectors
A vector in spherical polar co-ordinates is written as :
and
For spherical polar coordinate system we have h1=1, h2= r and h3= .
Fig 1.12(a) : Differential volume in s-p coordinates
15. Fig 1.12(b) : Exploded view
With reference to the Figure 1.12, the elemental areas are:
.......................(1.34)
and elementary volume is given by
........................(1.35)
16. Coordinate transformation between rectangular and spherical polar:
With reference to the figure 1.13 ,we can write the following equations:
........................................................(1.36)
Fig 1.13: Coordinate transformation
Given a vector in the spherical polar coordinate system, its component
in the cartesian coordinate system can be found out as follows
17. .................................(1.37)
Similarly,
.................................(1.38a)
.................................(1.38b)
The above equation can be put in a compact form:
.................................(1.39)
The components themselves will be functions of . are related
to x,y and z as:
....................(1.40)
and conversely,
.......................................(1.41a)
.................................(1.41b)
.....................................................(1.41c)
Using the variable transformation listed above, the vector components, which are functions of
variables of one coordinate system, can be transformed to functions of variables of other
coordinate system and a total transformation can be done.
Line, surface and volume integrals
In electromagnetic theory, we come across integrals, which contain vector functions. Some
representative integrals are listed below:
18. In the above integrals, and respectively represent vector and scalar function of space
coordinates. C,S and Vrepresent path, surface and volume of integration. All these integrals are
evaluated using extension of the usual one-dimensional integral as the limit of a sum, i.e., if a
function f(x) is defined over arrange a to b of values of x, then the integral is given by
.................................(1.42)
where the interval (a,b) is subdivided into n continuous interval of lengths .
Line Integral: Line integral is the dot product of a vector with a specified C; in other words
it is the integral of the tangential component along the curve C.
Fig 1.14: Line Integral
As shown in the figure 1.14, given a vector around C, we define the
integral as the line integral of E along the curve C.
19. If the path of integration is a closed path as shown in the figure the line integral becomes a closed
line integral and is called the circulation of around C and denoted as as shown in the
figure 1.15.
Fig 1.15: Closed Line Integral
Surface Integral :
Given a vector field , continuous in a region containing the smooth surface S, we define the
surface integral or the flux of through S as as surface
integral over surface S.
Fig 1.16 : Surface Integral
20. If the surface integral is carried out over a closed surface, then we write
Volume Integrals:
We define or as the volume integral of the scalar function f(function of spatial
coordinates) over the volume V. Evaluation of integral of the form can be carried out as a
sum of three scalar volume integrals, where each scalar volume integral is a component of the
vector
The Del Operator :
The vector differential operator was introduced by Sir W. R. Hamilton and later on developed
by P. G. Tait.
Mathematically the vector differential operator can be written in the general form as:
.................................(1.43)
In Cartesian coordinates:
................................................(1.44)
In cylindrical coordinates:
...........................................(1.45)
and in spherical polar coordinates:
.................................(1.46)
Gradient of a Scalar function:
Let us consider a scalar field V(u,v,w) , a function of space coordinates.
Gradient of the scalar field V is a vector that represents both the magnitude and direction of the maximum space
rate of increase of this scalar field V.
21. Fig 1.17 : Gradient of a scalar function
As shown in figure 1.17, let us consider two surfaces S1and S2 where the function V has constant
magnitude and the magnitude differs by a small amount dV. Now as one moves from S1 to S2, the
magnitude of spatial rate of change of Vi.e. dV/dl depends on the direction of elementary path
length dl, the maximum occurs when one traverses from S1toS2along a path normal to the
surfaces as in this case the distance is minimum
By our definition of gradient we can write:
.......................................................................(1.47)
since which represents the distance along the normal is the shortest distance between the
two surfaces.
For a general curvilinear coordinate system
....................(1.48)
Further we can write
......................................................(1.49)
Hence,
....................................(1.50)
22. Also we can write,
............................(1.51)
By comparison we can write,
....................................................................(1.52)
Hence for the Cartesian, cylindrical and spherical polar coordinate system, the expressions for
gradient can be written as:
In Cartesian coordinates:
...................................................................................(1.53)
In cylindrical coordinates:
..................................................................(1.54)
and in spherical polar coordinates:
..........................................................(1.55)
The following relationships hold for gradient operator.
...............................................................................(1.56)
where U and V are scalar functions and n is an integer.
23. It may further be noted that since magnitude of depends on the direction of dl, it is
called thedirectional derivative. If is called the scalar potential function of the
vector function .
Divergence of a Vector Field:
In study of vector fields, directed line segments, also called flux lines or streamlines, represent
field variations graphically. The intensity of the field is proportional to the density of lines. For
example, the number of flux lines passing through a unit surface S normal to the vector measures
the vector field strength.
Fig 1.18: Flux Lines
We have already defined flux of a vector field as
....................................................(1.57)
For a volume enclosed by a surface,
.........................................................................................(1.58)
We define the divergence of a vector field at a point P as the net outward flux from a volume
enclosing P, as the volume shrinks to zero.
.................................................................(1.59)
Here is the volume that encloses P and S is the corresponding closed surface.
24. Fig 1.19: Evaluation of divergence in curvilinear coordinate
Let us consider a differential volume centered on point P(u,v,w) in a vector field . The flux
through an elementary area normal to u is given by ,
........................................(1.60)
Net outward flux along u can be calculated considering the two elementary surfaces perpendicular to u .
.......................................(1.61)
Considering the contribution from all six surfaces that enclose the volume, we can write
.......................................(1.62)
Hence for the Cartesian, cylindrical and spherical polar coordinate system, the expressions for divergence can
be written as:
In Cartesian coordinates:
................................(1.63)
25. In cylindrical coordinates:
....................................................................(1.64)
and in spherical polar coordinates:
......................................(1.65)
In connection with the divergence of a vector field, the following can be noted
Divergence of a vector field gives a scalar.
..............................................................................(1.66)
Divergence theorem :
Divergence theorem states that the volume integral of the divergence of vector field is equal to
the net outward flux of the vector through the closed surface that bounds the volume.
Mathematically,
Proof:
Let us consider a volume V enclosed by a surface S . Let us subdivide the volume in large
number of cells. Let the kthcell has a volume and the corresponding surface is denoted by Sk.
Interior to the volume, cells have common surfaces. Outward flux through these common
surfaces from one cell becomes the inward flux for the neighboring cells. Therefore when the total
flux from these cells are considered, we actually get the net outward flux through the surface
surrounding the volume. Hence we can write:
......................................(1.67)
In the limit, that is when and the right hand of the expression can be written
as .
26. Hence we get , which is the divergence theorem.
Curl of a vector field:
We have defined the circulation of a vector field A around a closed path as .
Curl of a vector field is a measure of the vector field's tendency to rotate about a point. Curl ,
also written as is defined as a vector whose magnitude is maximum of the net circulation
per unit area when the area tends to zero and its direction is the normal direction to the area
when the area is oriented in such a way so as to make the circulation maximum.
Therefore, we can write:
......................................(1.68)
To derive the expression for curl in generalized curvilinear coordinate system, we first
compute and to do so let us consider the figure 1.20 :
If C1 represents the boundary of , then we can write
......................................(1.69)
The integrals on the RHS can be evaluated as follows:
.................................(1.70)
27. ................................................(1.71)
The negative sign is because of the fact that the direction of traversal reverses. Similarly,
..................................................(1.72)
............................................................................(1.73)
Adding the contribution from all components, we can write:
........................................................................(1.74)
Therefore,
......................................................(1.75)
In the same manner if we compute for and we can write,
.......(1.76)
This can be written as,
......................................................(1.77)
28. In Cartesian coordinates: .......................................(1.78)
In Cylindrical coordinates, ....................................(1.79)
In Spherical polar coordinates, ..............(1.80)
Curl operation exhibits the following properties:
..............(1.81)
Stoke's theorem :
It states that the circulation of a vector field around a closed path is equal to the integral
of over the surface bounded by this path. It may be noted that this equality holds
provided and are continuous on the surface.
i.e,
..............(1.82)
Proof:Let us consider an area S that is subdivided into large number of cells as shown in the
figure 1.21.
29. Fig 1.21: Stokes theorem
Let kthcell has surface area and is bounded path Lk while the total area is bounded by
path L. As seen from the figure that if we evaluate the sum of the line integrals around the
elementary areas, there is cancellation along every interior path and we are left the line integral
along path L. Therefore we can write,
..............(1.83)
As 0
. .............(1.84)
which is the stoke's theorem.
30. Solved Examples:
1. Given that
a. Determine the angle between the vectors A and B .
b. Find the unit vector which is perpendicular to both A and B .
Solution:
a. We know that for two given vector A and B,
For the two vectors A and B
or
b. We know that is perpendicular to both A and B.
=
The unit vector perpendicular to both A and B is given by,
31. 2. Given the vectors
Find :
a. The vector C = A + B at a point P (0, 2,-3).
b. The component of A along B at P.
Solution:
The vector B is cylindrical coordinates. This vector in Cartesian coordinate can be written as:
Where
The point P(0,2,-3) is in the y-z plane for which .
a. C = A + B
=
=
b. Component of A along B is where is the angle between A is and B.
32. i.e., =
3. A vector field is given by
Transform this vector into rectangular co-ordinates and calculate its magnitude at P(1,0,1).
Solution:
Given,
The components of the vector in Cartesian coordinates can be computed as follows:
33. 4. The coordinates of a point P in cylindrical co-ordinates is given by . Find the volume
of the sphere that has center at the origin and on which P is a point. If O represents the origin,
what angle OP subtends with z-axis?
Solution:
The radius of the sphere on which P is a point is given by
Therefore, the volume of the sphere