This document provides an introduction to electromagnetic theory, beginning with Maxwell's equations. It covers electrostatics, including Gauss' law and the Poisson equation. Magnetostatics, including Ampere's and Biot-Savart laws, are also discussed. The static scalar and vector potentials are introduced. Non-static fields, including electromagnetic waves and their propagation in conductors like RF cavities and waveguides, are then covered. Examples of allowed field patterns and modes in RF structures are shown.
This document provides preparatory notes and examples for an exam on electromagnetic theory. It covers key concepts like the Lorentz force equation, Biot-Savart law, Ampere's circuital law, Gauss's law for magnetism, and magnetic boundary conditions. Examples calculate the magnetic field and force on charges in various configurations like an infinite line current, parallel wires, and a ring of current. The document is a useful study guide summarizing the essential electromagnetic concepts and formulas tested on the exam.
The document discusses wave optics and electromagnetic waves. It defines key concepts like wavefronts, which connect points of equal phase, and rays, which describe the direction of wave propagation perpendicular to wavefronts. It explains Huygens' principle, which states that each point on a wavefront acts as a secondary source of spherical wavelets to determine the new wavefront position. The principle of superposition states that multiple waves add linearly at each point in space to determine the resulting disturbance. Interference occurs when waves are out of phase and their amplitudes diminish or vanish.
1 ECE 6340 Fall 2013 Homework 8 Assignment.docxjoyjonna282
1
ECE 6340
Fall 2013
Homework 8
Assignment: Please do Probs. 1-9 and 13 from the set below.
1) In dynamics, we have the equation
E j Aω= − −∇Φ .
(a) Show that in statics, the scalar potential function Φ can be interpreted as a voltage
function. That is, show that in statics
( ) ( )
B
AB
A
V E dr A B≡ ⋅ = Φ −Φ∫ .
(b) Next, explain why this equation is not true (in general) in dynamics.
(c) Explain why the voltage drop (defined as the line integral of the electric field, as
defined above) depends on the path from A to B in dynamics, using Faraday’s law.
(d) Does the right-hand side of the above equation (the difference in the potential
function) depend on the path, in dynamics?
Hint: Note that, according to calculus, for any function ψ we have
dr dx dy dz d
x y z
ψ ψ ψ
ψ ψ
∂ ∂ ∂
∇ ⋅ = + + =
∂ ∂ ∂
.
2) Starting with Maxwell’s equations, show that the electric field radiated by an impressed
current density source J i in an infinite homogeneous region satisfies the equation
( )2 2 iE k E E j Jωµ∇ + = ∇ ∇⋅ + .
Then use Ampere’s law (or, if you prefer, the continuity equation and the electric Gauss
law) to show that this equation may be written as
( )2 2 1 i iE k E J j J
j
ωµ
σ ωε
∇ + = − ∇ ∇⋅ +
+
.
2
Note that the total current density is the sum of the impressed current density and the
conduction current density, the latter obeying Ohm’s law (J c = σE).
Explain why this equation for the electric field would be harder to solve than the equation
that was derived in class for the magnetic vector potential.
3) Show that magnetic field radiated by an impressed current density source satisfies the
equation
2 2 iH k H J∇ + = −∇× .
Explain why this equation for the magnetic field would be harder to solve than the
equation that was derived in class for the magnetic vector potential.
4) Show that in a homogenous region of space the scalar electric potential satisfies the
equation
2 2
i
v
c
k
ρ
ε
∇ Φ + Φ = − ,
where ivρ is the impressed (source) charge density, which is the charge density that goes
along with the impressed current density, being related by
i ivJ jωρ∇⋅ = −
Hint: Start with E j Aω= − −∇Φ and take the divergence of both sides. Also, take the
divergence of both sides of Ampere’s law and use the continuity equation for the
impressed current (given above) to show that
1 ii v
c c
E J
j
ρ
ωε ε
∇⋅ = − ∇⋅ = .
Note: It is also true from the electric Gauss law that
vE
ρ
ε
∇⋅ = ,
but we prefer to have only an impressed (source) charge density on the right-hand side of
the equation for the potential Φ. In the time-harmonic steady state, assuming a
homogeneous and isotropic region, it follows that ρv = ρvi. That is, there is no charge
3
density arising from the conduction current. (If there were no impressed current sources,
the total charge density would therefore be ze ...
This document provides an overview of Maxwell's equations in free space and various coordinate systems. It discusses:
1) Maxwell's equations in differential and integral forms, including Gauss' law, Gauss' law for magnetism, Faraday's law, and Ampere's law.
2) The relationships between the differential and integral forms using theorems like the divergence theorem and Stokes' theorem.
3) How Maxwell's equations coupled the electric and magnetic fields and led to the prediction of electromagnetic waves traveling at the speed of light.
4) The equations of electrostatics, magnetostatics, electroquasistatics and magnetoquasistatics which describe situations where fields vary slowly or are time-
1) Maxwell's equations describe electromagnetic phenomena and relate electric and magnetic fields.
2) Charged particles move in curved paths due to electromagnetic fields, following the Lorentz force law. In a uniform magnetic field, particles follow helical trajectories with a characteristic gyrofrequency.
3) Electromagnetic waves propagate as oscillating electric and magnetic fields obeying the wave equation. Their speed in a vacuum is the speed of light.
This document discusses the quantization of electromagnetic radiation fields within the framework of quantum electrodynamics (QED). It begins by introducing the classical description of radiation fields in terms of vector potentials that satisfy the transversality condition. Next, it describes quantizing the classical radiation field by treating it as a collection of independent harmonic oscillators, with each oscillator characterized by a wave vector and polarization. Finally, it discusses how the quantization of these radiation oscillators leads to treating their canonical variables as non-commuting operators, in analogy to the quantization of position and momentum in non-relativistic quantum mechanics. This lays the foundation for a quantum description of radiation phenomena using the formalism of QED.
This lecture provides an overview of electromagnetic fields and Maxwell's equations. It introduces key concepts including electric and magnetic fields, Maxwell's equations in integral and differential form, electromagnetic boundary conditions, and electromagnetic fields in materials. Maxwell's equations are the fundamental laws of classical electromagnetics and govern all electromagnetic phenomena. The lecture also discusses phasor representation for time-harmonic fields.
This document discusses wave equations and their applications. It contains 6 chapters that cover basic results of wave equations, different types of wave equations like scalar and electromagnetic wave equations, and applications of wave equations. The introduction explains that the document contains details about different types of wave equations and their applications in sciences. Chapter 1 discusses basic concepts like relativistic and non-relativistic wave equations. Chapter 2 covers the scalar wave equation. Chapter 3 is about electromagnetic wave equations. Chapter 4 is on applications of wave equations.
This document provides preparatory notes and examples for an exam on electromagnetic theory. It covers key concepts like the Lorentz force equation, Biot-Savart law, Ampere's circuital law, Gauss's law for magnetism, and magnetic boundary conditions. Examples calculate the magnetic field and force on charges in various configurations like an infinite line current, parallel wires, and a ring of current. The document is a useful study guide summarizing the essential electromagnetic concepts and formulas tested on the exam.
The document discusses wave optics and electromagnetic waves. It defines key concepts like wavefronts, which connect points of equal phase, and rays, which describe the direction of wave propagation perpendicular to wavefronts. It explains Huygens' principle, which states that each point on a wavefront acts as a secondary source of spherical wavelets to determine the new wavefront position. The principle of superposition states that multiple waves add linearly at each point in space to determine the resulting disturbance. Interference occurs when waves are out of phase and their amplitudes diminish or vanish.
1 ECE 6340 Fall 2013 Homework 8 Assignment.docxjoyjonna282
1
ECE 6340
Fall 2013
Homework 8
Assignment: Please do Probs. 1-9 and 13 from the set below.
1) In dynamics, we have the equation
E j Aω= − −∇Φ .
(a) Show that in statics, the scalar potential function Φ can be interpreted as a voltage
function. That is, show that in statics
( ) ( )
B
AB
A
V E dr A B≡ ⋅ = Φ −Φ∫ .
(b) Next, explain why this equation is not true (in general) in dynamics.
(c) Explain why the voltage drop (defined as the line integral of the electric field, as
defined above) depends on the path from A to B in dynamics, using Faraday’s law.
(d) Does the right-hand side of the above equation (the difference in the potential
function) depend on the path, in dynamics?
Hint: Note that, according to calculus, for any function ψ we have
dr dx dy dz d
x y z
ψ ψ ψ
ψ ψ
∂ ∂ ∂
∇ ⋅ = + + =
∂ ∂ ∂
.
2) Starting with Maxwell’s equations, show that the electric field radiated by an impressed
current density source J i in an infinite homogeneous region satisfies the equation
( )2 2 iE k E E j Jωµ∇ + = ∇ ∇⋅ + .
Then use Ampere’s law (or, if you prefer, the continuity equation and the electric Gauss
law) to show that this equation may be written as
( )2 2 1 i iE k E J j J
j
ωµ
σ ωε
∇ + = − ∇ ∇⋅ +
+
.
2
Note that the total current density is the sum of the impressed current density and the
conduction current density, the latter obeying Ohm’s law (J c = σE).
Explain why this equation for the electric field would be harder to solve than the equation
that was derived in class for the magnetic vector potential.
3) Show that magnetic field radiated by an impressed current density source satisfies the
equation
2 2 iH k H J∇ + = −∇× .
Explain why this equation for the magnetic field would be harder to solve than the
equation that was derived in class for the magnetic vector potential.
4) Show that in a homogenous region of space the scalar electric potential satisfies the
equation
2 2
i
v
c
k
ρ
ε
∇ Φ + Φ = − ,
where ivρ is the impressed (source) charge density, which is the charge density that goes
along with the impressed current density, being related by
i ivJ jωρ∇⋅ = −
Hint: Start with E j Aω= − −∇Φ and take the divergence of both sides. Also, take the
divergence of both sides of Ampere’s law and use the continuity equation for the
impressed current (given above) to show that
1 ii v
c c
E J
j
ρ
ωε ε
∇⋅ = − ∇⋅ = .
Note: It is also true from the electric Gauss law that
vE
ρ
ε
∇⋅ = ,
but we prefer to have only an impressed (source) charge density on the right-hand side of
the equation for the potential Φ. In the time-harmonic steady state, assuming a
homogeneous and isotropic region, it follows that ρv = ρvi. That is, there is no charge
3
density arising from the conduction current. (If there were no impressed current sources,
the total charge density would therefore be ze ...
This document provides an overview of Maxwell's equations in free space and various coordinate systems. It discusses:
1) Maxwell's equations in differential and integral forms, including Gauss' law, Gauss' law for magnetism, Faraday's law, and Ampere's law.
2) The relationships between the differential and integral forms using theorems like the divergence theorem and Stokes' theorem.
3) How Maxwell's equations coupled the electric and magnetic fields and led to the prediction of electromagnetic waves traveling at the speed of light.
4) The equations of electrostatics, magnetostatics, electroquasistatics and magnetoquasistatics which describe situations where fields vary slowly or are time-
1) Maxwell's equations describe electromagnetic phenomena and relate electric and magnetic fields.
2) Charged particles move in curved paths due to electromagnetic fields, following the Lorentz force law. In a uniform magnetic field, particles follow helical trajectories with a characteristic gyrofrequency.
3) Electromagnetic waves propagate as oscillating electric and magnetic fields obeying the wave equation. Their speed in a vacuum is the speed of light.
This document discusses the quantization of electromagnetic radiation fields within the framework of quantum electrodynamics (QED). It begins by introducing the classical description of radiation fields in terms of vector potentials that satisfy the transversality condition. Next, it describes quantizing the classical radiation field by treating it as a collection of independent harmonic oscillators, with each oscillator characterized by a wave vector and polarization. Finally, it discusses how the quantization of these radiation oscillators leads to treating their canonical variables as non-commuting operators, in analogy to the quantization of position and momentum in non-relativistic quantum mechanics. This lays the foundation for a quantum description of radiation phenomena using the formalism of QED.
This lecture provides an overview of electromagnetic fields and Maxwell's equations. It introduces key concepts including electric and magnetic fields, Maxwell's equations in integral and differential form, electromagnetic boundary conditions, and electromagnetic fields in materials. Maxwell's equations are the fundamental laws of classical electromagnetics and govern all electromagnetic phenomena. The lecture also discusses phasor representation for time-harmonic fields.
This document discusses wave equations and their applications. It contains 6 chapters that cover basic results of wave equations, different types of wave equations like scalar and electromagnetic wave equations, and applications of wave equations. The introduction explains that the document contains details about different types of wave equations and their applications in sciences. Chapter 1 discusses basic concepts like relativistic and non-relativistic wave equations. Chapter 2 covers the scalar wave equation. Chapter 3 is about electromagnetic wave equations. Chapter 4 is on applications of wave equations.
This document provides an overview of electromagnetics and Maxwell's equations. It includes:
1. A table of contents listing key topics such as conservation of charges, electromagnetic wave equation, boundary conditions, and energy and momentum.
2. Background on Maxwell's equations and the key scientists involved in developing an understanding of electromagnetism, including Coulomb, Ørsted, Ampère, Biot, Savart, and Faraday.
3. Expressions of Maxwell's equations and the electromagnetic field quantities they relate such as electric field, magnetic field, electric displacement, magnetic induction, and current density.
4. Discussion of important concepts that can be derived from Maxwell's equations, including the
Fermi surface and de haas van alphen effect ppttedoado
The document discusses the Onsager theory of semiclassical quantization of electron orbits in a magnetic field. It describes how the Bohr-Sommerfeld quantization condition leads to the quantization of the magnetic flux through an electron orbit. This quantization of flux results in discrete Landau levels with energies dependent on the quantum number and magnetic field strength. Measurements of oscillations in magnetization via the de Haas-van Alphen effect can be used to map the Fermi surface by detecting extremal orbits corresponding to peaks in the density of states.
- The document discusses magnetic fields created by electric currents. It covers the magnetic field of a moving point charge, the Biot-Savart law for calculating the magnetic field from a current-carrying wire, and an example calculation of the magnetic field from a long straight wire.
- The right hand rule is introduced for determining the direction of magnetic fields.
- Maxwell's equations for static magnetic fields in integral and differential form are presented.
- The document discusses magnetic fields created by electric currents. It covers the magnetic field of a moving point charge, the Biot-Savart law for calculating the magnetic field from a current-carrying wire, and an example calculation of the magnetic field from a long straight wire.
- The right hand rule is introduced for determining the direction of magnetic fields.
- Maxwell's equations for static magnetic fields in integral and differential form are presented.
Maxwell's Equations describe the fundamental interactions between electric and magnetic fields. They consist of four equations:
1) Gauss's law relates electric charge density to electric field.
2) Gauss's law for magnetism states that magnetic monopoles have not been observed.
3) Faraday's law describes how a changing magnetic field induces an electric field.
4) Ampere-Maxwell law relates electric current and changing electric fields to magnetic fields.
Together, Maxwell's Equations show that changing electric and magnetic fields propagate as electromagnetic waves moving at the speed of light.
Maxwell's Equations describe the fundamental interactions between electric and magnetic fields. They consist of four equations:
1) Gauss's law relates electric charge density to electric field.
2) Gauss's law for magnetism states that magnetic monopoles have not been observed.
3) Faraday's law describes how a changing magnetic field induces an electric field.
4) Ampere-Maxwell law relates electric current and changing electric fields to magnetic fields.
Together, these equations show that changing electric and magnetic fields propagate as electromagnetic waves travelling at the speed of light.
Maxwell's equations unified electricity, magnetism, and light by showing that electromagnetic waves propagate through space at a speed c. The equations predicted that changing electric and magnetic fields produce transverse waves that transport energy and momentum. Maxwell's work established that light is an electromagnetic wave oscillating perpendicular to the direction of propagation.
1) Maxwell's equations describe electromagnetic waves propagating through space and time. For time-varying fields, the full set of Maxwell's equations must be used.
2) By assuming time-harmonic fields with a sinusoidal time variation, Maxwell's equations can be simplified to phasor forms containing only spatial derivatives.
3) The phasor forms of Maxwell's equations can be reduced to Helmholtz wave equations for the electric and magnetic fields. Plane wave solutions representing uniform electromagnetic waves propagating in a given direction can be derived from these equations.
Maxwell's equations and their derivations.Praveen Vaidya
Being the partial differential equations along with the Lorentz law the Maxwell's equation laid the foundation for classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.[note 1] One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the "speed of light". Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.
I am Irene M. I am an Electromagnetism Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Electromagnetism, from California, USA. I have been helping students with their homework for the past 8 years. I solve assignments related to Electromagnetism.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Electromagnetism Assignments.
1. The document discusses electric flux and Gauss's law. It defines electric flux as the measure of the strength of the electric field penetrating a surface, and gives the formula for calculating electric flux.
2. Gauss's law states that the net electric flux through a closed surface is equal to the electric charge enclosed divided by the permittivity of free space. The document provides examples of calculating electric flux and applying Gauss's law.
3. Key concepts covered include the relationship between electric field and flux, the effect of angle between field and area on flux, and using Gauss's law to determine flux through a surface based on enclosed charge.
1) James Clerk Maxwell unified existing laws of electricity and magnetism through his equations, revealing that changing electric and magnetic fields propagate as electromagnetic waves traveling at the speed of light.
2) Solving Maxwell's equations results in the wave equation, showing that light is an electromagnetic wave.
3) Electromagnetic waves carry energy through space, and all remote sensing is based on the modulation of this energy.
This document provides an introduction to quantizing the electromagnetic field. It begins with a classical description of the electromagnetic field using Maxwell's equations. It then shows that the classical electromagnetic field can be described as an infinite collection of independent harmonic oscillators. The document proceeds to quantize these harmonic oscillators by promoting the classical variables to quantum operators. This leads to a description of the electromagnetic field in terms of photon creation and annihilation operators. The quantized electromagnetic field gives rise to phenomena like zero-point energy and the Casimir effect that cannot be explained classically.
Gauss's law relates the electric flux through a closed surface to the enclosed electric charge. The total flux is equal to the enclosed charge divided by the permittivity of free space. Gauss's law can be used to easily calculate the electric field due to highly symmetric charge distributions, such as a point charge. While Gauss's law and Coulomb's law are equivalent, Gauss's law provides a convenient approach for some problems due to its emphasis on the total electric flux.
Zero Point Energy And Vacuum Fluctuations EffectsAna_T
The document discusses several topics related to zero point energy and vacuum fluctuations:
1) It explains how the Heisenberg uncertainty principle leads to zero point energy, which is the lowest possible energy of a quantum system in its ground state.
2) It describes vacuum fluctuations as quantum fluctuations of fields even in their lowest energy state. These fluctuations can be thought of as virtual particles being created and destroyed in the vacuum.
3) Several phenomena are discussed that demonstrate the effects of zero point energy and vacuum fluctuations, including spontaneous emission, the Casimir effect, and the Lamb shift.
1) The document provides one mark, two mark and three mark questions from the chapter on Electric Charges and Fields.
2) It includes questions testing definitions of key terms like electric charge, electric field, electric dipole moment, Gauss's law.
3) It also has questions requiring diagrams of electric field patterns and derivations of expressions for force between charges and electric field.
1) The document provides one mark, two mark and three mark questions from the chapter on Electric Charges and Fields.
2) It includes questions testing definitions of key terms like electric charge, electric field, electric dipole moment, Gauss's law.
3) It also has questions requiring diagrams of electric field patterns and derivations of expressions for force between charges and electric field.
This document provides an overview of electromagnetics and Maxwell's equations. It includes:
1. A table of contents listing key topics such as conservation of charges, electromagnetic wave equation, boundary conditions, and energy and momentum.
2. Background on Maxwell's equations and the key scientists involved in developing an understanding of electromagnetism, including Coulomb, Ørsted, Ampère, Biot, Savart, and Faraday.
3. Expressions of Maxwell's equations and the electromagnetic field quantities they relate such as electric field, magnetic field, electric displacement, magnetic induction, and current density.
4. Discussion of important concepts that can be derived from Maxwell's equations, including the
Fermi surface and de haas van alphen effect ppttedoado
The document discusses the Onsager theory of semiclassical quantization of electron orbits in a magnetic field. It describes how the Bohr-Sommerfeld quantization condition leads to the quantization of the magnetic flux through an electron orbit. This quantization of flux results in discrete Landau levels with energies dependent on the quantum number and magnetic field strength. Measurements of oscillations in magnetization via the de Haas-van Alphen effect can be used to map the Fermi surface by detecting extremal orbits corresponding to peaks in the density of states.
- The document discusses magnetic fields created by electric currents. It covers the magnetic field of a moving point charge, the Biot-Savart law for calculating the magnetic field from a current-carrying wire, and an example calculation of the magnetic field from a long straight wire.
- The right hand rule is introduced for determining the direction of magnetic fields.
- Maxwell's equations for static magnetic fields in integral and differential form are presented.
- The document discusses magnetic fields created by electric currents. It covers the magnetic field of a moving point charge, the Biot-Savart law for calculating the magnetic field from a current-carrying wire, and an example calculation of the magnetic field from a long straight wire.
- The right hand rule is introduced for determining the direction of magnetic fields.
- Maxwell's equations for static magnetic fields in integral and differential form are presented.
Maxwell's Equations describe the fundamental interactions between electric and magnetic fields. They consist of four equations:
1) Gauss's law relates electric charge density to electric field.
2) Gauss's law for magnetism states that magnetic monopoles have not been observed.
3) Faraday's law describes how a changing magnetic field induces an electric field.
4) Ampere-Maxwell law relates electric current and changing electric fields to magnetic fields.
Together, Maxwell's Equations show that changing electric and magnetic fields propagate as electromagnetic waves moving at the speed of light.
Maxwell's Equations describe the fundamental interactions between electric and magnetic fields. They consist of four equations:
1) Gauss's law relates electric charge density to electric field.
2) Gauss's law for magnetism states that magnetic monopoles have not been observed.
3) Faraday's law describes how a changing magnetic field induces an electric field.
4) Ampere-Maxwell law relates electric current and changing electric fields to magnetic fields.
Together, these equations show that changing electric and magnetic fields propagate as electromagnetic waves travelling at the speed of light.
Maxwell's equations unified electricity, magnetism, and light by showing that electromagnetic waves propagate through space at a speed c. The equations predicted that changing electric and magnetic fields produce transverse waves that transport energy and momentum. Maxwell's work established that light is an electromagnetic wave oscillating perpendicular to the direction of propagation.
1) Maxwell's equations describe electromagnetic waves propagating through space and time. For time-varying fields, the full set of Maxwell's equations must be used.
2) By assuming time-harmonic fields with a sinusoidal time variation, Maxwell's equations can be simplified to phasor forms containing only spatial derivatives.
3) The phasor forms of Maxwell's equations can be reduced to Helmholtz wave equations for the electric and magnetic fields. Plane wave solutions representing uniform electromagnetic waves propagating in a given direction can be derived from these equations.
Maxwell's equations and their derivations.Praveen Vaidya
Being the partial differential equations along with the Lorentz law the Maxwell's equation laid the foundation for classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.[note 1] One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the "speed of light". Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.
I am Irene M. I am an Electromagnetism Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Electromagnetism, from California, USA. I have been helping students with their homework for the past 8 years. I solve assignments related to Electromagnetism.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Electromagnetism Assignments.
1. The document discusses electric flux and Gauss's law. It defines electric flux as the measure of the strength of the electric field penetrating a surface, and gives the formula for calculating electric flux.
2. Gauss's law states that the net electric flux through a closed surface is equal to the electric charge enclosed divided by the permittivity of free space. The document provides examples of calculating electric flux and applying Gauss's law.
3. Key concepts covered include the relationship between electric field and flux, the effect of angle between field and area on flux, and using Gauss's law to determine flux through a surface based on enclosed charge.
1) James Clerk Maxwell unified existing laws of electricity and magnetism through his equations, revealing that changing electric and magnetic fields propagate as electromagnetic waves traveling at the speed of light.
2) Solving Maxwell's equations results in the wave equation, showing that light is an electromagnetic wave.
3) Electromagnetic waves carry energy through space, and all remote sensing is based on the modulation of this energy.
This document provides an introduction to quantizing the electromagnetic field. It begins with a classical description of the electromagnetic field using Maxwell's equations. It then shows that the classical electromagnetic field can be described as an infinite collection of independent harmonic oscillators. The document proceeds to quantize these harmonic oscillators by promoting the classical variables to quantum operators. This leads to a description of the electromagnetic field in terms of photon creation and annihilation operators. The quantized electromagnetic field gives rise to phenomena like zero-point energy and the Casimir effect that cannot be explained classically.
Gauss's law relates the electric flux through a closed surface to the enclosed electric charge. The total flux is equal to the enclosed charge divided by the permittivity of free space. Gauss's law can be used to easily calculate the electric field due to highly symmetric charge distributions, such as a point charge. While Gauss's law and Coulomb's law are equivalent, Gauss's law provides a convenient approach for some problems due to its emphasis on the total electric flux.
Zero Point Energy And Vacuum Fluctuations EffectsAna_T
The document discusses several topics related to zero point energy and vacuum fluctuations:
1) It explains how the Heisenberg uncertainty principle leads to zero point energy, which is the lowest possible energy of a quantum system in its ground state.
2) It describes vacuum fluctuations as quantum fluctuations of fields even in their lowest energy state. These fluctuations can be thought of as virtual particles being created and destroyed in the vacuum.
3) Several phenomena are discussed that demonstrate the effects of zero point energy and vacuum fluctuations, including spontaneous emission, the Casimir effect, and the Lamb shift.
1) The document provides one mark, two mark and three mark questions from the chapter on Electric Charges and Fields.
2) It includes questions testing definitions of key terms like electric charge, electric field, electric dipole moment, Gauss's law.
3) It also has questions requiring diagrams of electric field patterns and derivations of expressions for force between charges and electric field.
1) The document provides one mark, two mark and three mark questions from the chapter on Electric Charges and Fields.
2) It includes questions testing definitions of key terms like electric charge, electric field, electric dipole moment, Gauss's law.
3) It also has questions requiring diagrams of electric field patterns and derivations of expressions for force between charges and electric field.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
ACRP 4-09 Risk Assessment Method to Support Modification of Airfield Separat...
Lectures.pdf
1. Introduction to the Electromagnetic Theory
Andrea Latina (CERN)
andrea.latina@cern.ch
Basics of Accelerator Physics and Technology
7-11 October 2019, Archamps, France
2. Table of contents
I Introduction
I Electrostatics
I E.g. Space-charge forces
I Magnetostatics
I E.g. Accelerator magnets
I Non-static case
I E.g. RF acceleration and wave guides
2/37 A. Latina - Electromagnetic Theory
4. Motivation: control of charged particle beams
To control a charged particle beam we use electromagnetic fields. Recall the
Lorentz force:
~
F = q ·
~
E + ~
v × ~
B
where, in high energy machines, |~
v| ≈ c ≈ 3 · 108
m/s. In particle accelerators,
transverse deflection is usually given by magnetic fields, whereas acceleration can
only be given by electric fields.
Comparison of electric and magnetic force:
~
E = 1 MV/m
~
B = 1 T
Fmagnetic
Felectric
=
evB
eE
=
βcB
E
' β
3 · 108
106
= 300 β
⇒ the magnetic force is much stronger then the electric one: in an accelerator, use
magnetic fields whenever possible.
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5. Some references
1. Richard P. Feynman, Lectures on Physics, 1963, on-line
2. J. D. Jackson, Classical Electrodynamics, Wiley, 1998
3. David J. Griffiths, Introduction to Electrodynamics, Cambridge University Press, 2017
4. Thomas P. Wangler, RF Linear Accelerators, Wiley, 2008
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6. Variables and units
E electric field [V/m]
B magnetic field [T]
D electric displacement [C/m2
]
H magnetizing field [A/m]
q electric charge [C]
ρ electric charge density [C/m3
]
j = ρv current density [A/m2
]
0 permittivity of vacuum, 8.854 · 10−12
[F/m]
µ0 = 1
0c2 permeability of vacuum, 4π · 10−7
[H/m or N/A2
]
c speed of light in vacuum, 2.99792458 · 108
[m/s]
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7. Differentiation with vectors
I We define the operator “nabla”:
∇
def
=
∂
∂x , ∂
∂y , ∂
∂z
which we treat as a special vector.
I Examples:
∇ · F =
∂Fx
∂x
+
∂Fy
∂y
+
∂Fz
∂z
divergence
∇ × F =
∂Fz
∂y − ∂Fy
∂z , ∂Fx
∂z − ∂Fz
∂x , ∂Fy
∂x − ∂Fx
∂y
curl
∇φ =
∂φ
∂x , ∂φ
∂y , ∂φ
∂z
gradient
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8. Maxwell’s equations: integral form
1. Maxwell’s equations can be written in integral or in differential form (SI
units convention):
(1) Gauss’ law;
(2) Gauss’ law for magnetism;
(3) Maxwell–Faraday equation (Faraday’s law of induction);
(4) Ampère’s circuital law
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9. Maxwell’s equations: differential form
1. Maxwell’s equations can be written in integral or in differential form (SI
units convention):
(1) Gauss’ law;
(2) Gauss’ law for magnetism;
(3) Maxwell–Faraday equation (Faraday’s law of induction);
(4) Ampère’s circuital law
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11. Static case
I We will consider relatively simple situations.
I The easiest circumstance is one in which nothing depends on the time—this is
called the static case:
I All charges are permanently fixed in space, or if they do move, they move as
a steady flow in a circuit (so ρ and j are constant in time).
I In these circumstances, all of the terms in the Maxwell equations which are time
derivatives of the field are zero. In this case, the Maxwell equations become:
11/37 A. Latina - Electromagnetic Theory
12. Electrostatics: principle of superposition
I Coulomb’s Law: Electric field due to a stationary point charge q, located in r1:
E (r) =
q
4π0
r − r1
|r − r1|3
I Principle of superposition, tells that a distribution of charges qi generates an
electric field:
E (r) =
1
4π0
X
qi
r − ri
|r − ri |3
I Continuous distribution of charges, ρ (r)
E (r) =
1
4π0
˚
V
ρ r0 r − r0
|r − r0|3
dr
with Q =
˝
V
ρ (r0
) dr as the total charge, and where ρ is the charge density.
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14. Electrostatics: Gauss’ law
Gauss’ law states that the flux of ~
E is:
¨
A
~
E·d~
A=
ˆ
any closed
surface A
En da=
sum of charges inside A
0
We know that
E (r) =
1
4π0
˚
V
ρ r0
r − r0
|r − r0|3
dr
In differential form, using the Gauss’ theorem (diver-
gence theorem):
¨
~
E · d~
A =
˚
∇ · ~
E dr
which gives the first Maxwell’s equation in differential
form:
∇ · ~
E=
ρ
0
Example: case of a single point charge
¨
~
E · d~
A =
(
q
0
if q lies inside A
0 if q lies outside A
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16. Electrostatics: scalar potential and Poisson equation
The equations for electrostatics are:
∇ · ~
E =
ρ
0
∇ × ~
E = 0
The two can be combined into a single equation:
~
E=-∇φ
which leads to the Poisson’s equation:
∇ · ∇φ = ∇2
φ=-
ρ
0
Where the operator ∇2 is called Laplacian:
∇ · ∇ = ∇2
=
∂2
∂x2
+
∂2
∂y2
+
∂2
∂z2
The Poisson’s equation allows to compute the electric field generated by arbitrary
charge distributions.
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19. Magnetostatics: Ampère’s and Biot-Savart laws
The equations for electrostatics are:
∇ · ~
B = 0
∇ × ~
B =
~
j
0c2
The Stokes’ theorem tells that:
˛
C
~
B · d~
r =
¨
A
∇ × ~
B
· d~
A
This equation gives the Ampère’s law:
˛
C
~
B · d~
r =
1
0c2
¨
A
~
j · ~
n dA
From which one can derive the Biot-Savart law, stating that, along a current j:
~
B (~
r) =
1
4π0c2
=
˛
C
j d~
r0
× (~
r −~
r0
)
|~
r −~
r0|3
This provides a practical way to compute ~
B from current distributions.
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20. Magnetostatics: vector potential
The equations for electrostatics are:
∇ · ~
B = 0
∇ × ~
B =
~
j
0c2
They can be unified into one, introducing the vector potential ~
A:
~
B = ∇ × ~
A
Using the Stokes’ theorem
~
B (~
r) =
1
4π0c2
=
˛
C
j d~
r0
× (~
r −~
r0
)
|~
r −~
r0|3
one can derive the expression of the vector potential ~
A from of the current ~
j:
~
A (r) =
µ0
4π
˚ ~
j (~
r0
)
|~
r −~
r0|
d3
~
r0
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21. Summary of electro- and magneto- statics
One can compute the electric and the magnetic fields from the scalar and the vector
potentials
~
E = −∇φ
~
B = ∇ × ~
A
with
φ (r) =
1
4π0
˚
ρ (~
r0
)
|~
r −~
r0|
d3
~
r0
~
A (r) =
µ0
4π
˚ ~
j (~
r0
)
|~
r −~
r0|
d3
~
r0
being ρ the charge density, and ~
j the current density.
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22. Motion of a charged particle in an electric field
~
F = q ·
~
E + ~
v ×
S
S
~
B
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23. Motion of a charged particle in a magnetic field
~
F = q ·
S
S
~
E + ~
v × ~
B
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25. Magnetostatics: Faraday’s law of induction
“The electromotive force around a closed path is equal to the negative of the time rate
of change of the magnetic flux enclosed by the path.”
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26. Non-static case: electromagnetic waves
Electromagnetic wave equation:
~
E (~
r, t) = ~
E0ei(ωt−~
k·~
r)
~
B (~
r, t) = ~
B0ei(ωt−~
k·~
r)
Important quantities:
~
k =
2π
λ
=
ω
c
wave-number vector
λ =
c
f
wave length
f frequency
ω = 2πf angular frequency
Magnetic and electric fields are transverse to direction of propagation:
~
E ⊥ ~
B ⊥ ~
k
Short wave length →high frequency → high energy
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27. Spectrum of electromagnetic waves
Examples:
I yellow light ≈ 5 · 1014
Hz (i.e. ≈ 2 eV !)
I LEP (SR) ≤ 2 · 1020
Hz (i.e. ≈ 0.8 MeV !)
I gamma rays ≤ 3 · 1021
Hz (i.e. ≤ 12 MeV !)
(For estimates using temperature: 3 K ≈ 0.00025 eV )
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28. Electromagnetic waves impacting highly conductive
materials
Highly conductive materials: RF cavities, wave guides.
I In an ideal conductor:
~
Ek = 0, ~
B⊥ = 0
I This implies:
I All energy of an electromagnetic wave is reflected from the surface of an
ideal conductor.
I Fields at any point in the ideal conductor are zero.
I Only some field patterns are allowed in waveguides and RF cavities.
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37. ...The End!
Thank you
for your attention!
Special thanks to Werner Herr, for the pictures I took from his slides.
37/37 A. Latina - Electromagnetic Theory