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-
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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.
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
Maxwell equations without a polarization field august 15 1 2020Bob Eisenberg
Electrodynamics is almost always written using a polarization vector field to describe the response of matter to an electric field, or more specifically, to describe the change in distribution of charges as an electric field is applied or changed. This approach does not allow unique specification of a polarization field from measurements of the electric and magnetic fields and electrical current.
Many polarization fields will produce the same electric and magnetic fields, and current, because only the divergence of the polarization enters Maxwell’s first equation, relating charge and electric field. The curl of any function can be added to a polarization field without changing the electric field at all. The divergence of the curl is always zero. Models of structures that produce polarization cannot be uniquely determined from electrical measurements for the same reason. Models must describe charge distribution not just distribution of polarization to be unique.
I propose a different approach, using a different paradigm to describe field dependent charge, i.e., to describe the phenomena of polarization. I propose an operational definition of polarization that has worked well in biophysics where a field dependent, time dependent polarization provides the gating current that makes neuronal sodium and potassium channels respond to voltage. The operational definition has been applied successfully to experiments for nearly fifty years. Estimates of polarization have been computed from simulations, models, and theories using this definition and they fit experimental data quite well.
I propose that the same operational definition be used to define polarization charge in experiments, models, computations, theories, and simulations of other systems. Charge movement needs to be computed from a combination of electrodynamics and mechanics because ‘everything interacts with everything else’.
The classical polarization field need not enter into that treatment at all.
Describes the general solutions of Electromagnetic Maxwell Equations.
Intended or Graduate Students in Science (math, physics, engineering) with previous knowledge in electromagnetics.
Please send me comments and suggestions for improvements to solo.hermelin@gmail.com.
More presentations can be found in my website at http://www.solohermelin.com.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
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.
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
Maxwell equations without a polarization field august 15 1 2020Bob Eisenberg
Electrodynamics is almost always written using a polarization vector field to describe the response of matter to an electric field, or more specifically, to describe the change in distribution of charges as an electric field is applied or changed. This approach does not allow unique specification of a polarization field from measurements of the electric and magnetic fields and electrical current.
Many polarization fields will produce the same electric and magnetic fields, and current, because only the divergence of the polarization enters Maxwell’s first equation, relating charge and electric field. The curl of any function can be added to a polarization field without changing the electric field at all. The divergence of the curl is always zero. Models of structures that produce polarization cannot be uniquely determined from electrical measurements for the same reason. Models must describe charge distribution not just distribution of polarization to be unique.
I propose a different approach, using a different paradigm to describe field dependent charge, i.e., to describe the phenomena of polarization. I propose an operational definition of polarization that has worked well in biophysics where a field dependent, time dependent polarization provides the gating current that makes neuronal sodium and potassium channels respond to voltage. The operational definition has been applied successfully to experiments for nearly fifty years. Estimates of polarization have been computed from simulations, models, and theories using this definition and they fit experimental data quite well.
I propose that the same operational definition be used to define polarization charge in experiments, models, computations, theories, and simulations of other systems. Charge movement needs to be computed from a combination of electrodynamics and mechanics because ‘everything interacts with everything else’.
The classical polarization field need not enter into that treatment at all.
Describes the general solutions of Electromagnetic Maxwell Equations.
Intended or Graduate Students in Science (math, physics, engineering) with previous knowledge in electromagnetics.
Please send me comments and suggestions for improvements to solo.hermelin@gmail.com.
More presentations can be found in my website at http://www.solohermelin.com.
Intuitive explanation of maxwell electromagnetic equationsAbdiasis Jama
user friendly explanation of maxwell equations
For complete planning and design of microwave and cellular system, get this new book from Amazon
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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.
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You can also call on +1 678 648 4277 for any assistance with Electromagnetism Assignments.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
1. 1
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Lecture 2
Maxwell’s Equations in Free Space
In this lecture you will learn:
• Co-ordinate Systems and Course Notations
• Maxwell’s Equations in Differential and Integral Forms
• Electrostatics and Magnetostatics
• Electroquasistatics and Magnetoquasistatics
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Co-ordinate Systems and Vectors
Cartesian Coordinate System
zAyAxAA zyx ˆˆˆ ++=
r
kAjAiAA zyx
ˆˆˆ ++=
r
zzyyxx iAiAiAA ˆˆˆ ++=
r
All mean exactly the same thing …
just a different notation for the unit
vectors
The first one will be used in this
course
y
ox
z
y
x
z
oz
( )ooo zyx ,,
oy
Vectors in Cartesian Coordinate System
2. 2
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Co-ordinate Systems and Vectors
Cylindrical Coordinate System
zAArAA zr ˆˆˆ ++= φφ
r
zzrr iAiAiAA ˆˆˆ ++= φφ
r
y
x
z
oφ
or oz
( )ooo zr ,,φ
Vectors in Cylindrical Coordinate System
Both mean exactly the same thing …
just a different notation for the unit
vectors
The first one will be used in this
course
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Co-ordinate Systems and Vectors
Spherical Coordinate System
θφ θφ
ˆˆˆ AArAA r ++=
r
θθφφ iAiAiAA rr
ˆˆˆ ++=
r
y
x
z
oφ
or
( )ooor θφ ,,
Vectors in Spherical Coordinate System
oθ
Both mean exactly the same thing …
just a different notation for the unit
vectors
The first one will be used in this
course
3. 3
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Vector Fields
In layman terms, a vector field implies a vector associated with every point is
space:
Examples:
Electric Field:
Magnetic Field:
( ) ( )trEtzyxE ,or,,,
rrr
( ) ( )trHtzyxH ,or,,,
rrr
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Maxwell’s Equations in Free Space
t
E
JH
t
H
E
H
E
o
o
o
o
∂
∂
+=×∇
∂
∂
−=×∇
=∇
=∇
r
rr
r
r
r
r
ε
µ
µ
ρε
0.
.
adE
t
adJsdH
adH
t
sdE
adH
dVadE
o
o
o
o
rrrrrr
rrrr
rr
rr
...)4(
..)3(
0.)2(
.)1(
∫∫
∂
∂
+∫∫=∫
∫∫
∂
∂
−=∫
=∫∫
∫∫∫=∫∫
ε
µ
µ
ρε Gauss’ Law
Gauss’ Law
Faraday’s Law
Ampere’s Law
Integral Form Differential Form
Lorentz Force Law
( )HvEqF o
rrrr
µ×+=
Lorentz Law describes the effect of
electromagnetic fields upon charges
4. 4
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Physical Quantities, Values, and SI Units
E
r
Electric Field Volts/m
Magnetic Field Amps/m
Permittivity of Free Space 8.85x10-12 Farads/m
Permeability of Free Space 4πx10-7 Henry/m
Electronic Unit of Charge 1.6x10-19 Coulombs
Volume Charge Density Coulombs/m3
Current Density Amps/m2
Electric Flux Density Coulombs/m2
Magnetic Flux Density Tesla
Quantity
H
r
oε
oµ
q
ρ
ED o
rr
ε=
HB o
rr
µ=
J
r
Value/Units
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Gauss’ Law – Integral Form
dVadDdVadEo ∫∫∫=∫∫∫∫∫=∫∫ ρρε
rrrr
.or.
What is this law saying ……??
( )zyx ,,ρ
A closed surface of
arbitrary shape
surrounding a
charge distribution
Gauss’ Law: The total electric flux coming out of a closed surface is equal to the
total charge enclosed by that closed surface (irrespective of the shape of the
closed surface)
Points to Note: This law establishes charges as the “sources” or “sinks” of the
electric field (i.e. charges produce or terminate electric field lines).
If the total flux through a closed surface is positive, then the total charge enclosed
is positive. If the total flux is negative, then the total charge enclosed is negative
Carl F. Gauss
(1777-1855)
D
5. 5
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Gauss’ Law – Differential Form
dVAadA ∫∫∫ ∇=∫∫
rrr
..
Divergence Theorem:
For any vector field:
The flux of a vector through a closed surface is equal to the integral of the
divergence of the vector taken over the volume enclosed by that closed
surface
Using the Divergence Theorem with Gauss’ Law in Integral Form:
ρερ
ρ
ρ
=∇=∇⇒
∫∫∫=∫∫∫ ∇⇒
∫∫∫=∫∫
ED
dVdVD
dVadD
o
rr
r
rr
.or.
.
.
( )zyx ,,ρ
D
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Gauss’ Law for the Magnetic Field – Integral Form
0.or0. =∫∫=∫∫ adBadHo
rrrr
µ
What is this law saying ……?? A closed surface of
arbitrary shape
Gauss’ Law for the Magnetic Fields: The total magnetic flux coming out of a
closed surface is always zero.
Points to Note: This law implies that there are no such things as “magnetic
charges” that can emanate or terminate magnetic field lines.
If magnetic field is non-zero, then the flux into any closed surface must equal the
flux out of it - so that the net flux coming out is zero.
B
6. 6
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Gauss’ Law – Differential Form
dVAadA ∫∫∫ ∇=∫∫
rrr
..
Divergence Theorem:
For any vector field:
The flux of a vector through a closed surface is equal to the
integral of the divergence of the vector taken over the volume
enclosed by that closed surface
Using the Divergence Theorem with Gauss’ Law for the Magnetic Field in
Integral form:
0.or0.
0.
)(Remember0.
=∇=∇⇒
=∫∫∫ ∇⇒
==∫∫
HB
dVB
HBadB
o
o
rr
r
rrrr
µ
µ
B
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Faraday’s Law – Integral Form
adB
t
sdEadH
t
sdE o
rrrrrrrr
..or.. ∫∫
∂
∂
−=∫∫∫
∂
∂
−=∫ µ
What is this law saying ……??
A closed contour
Faraday’s Law: The line integral of electric field over a closed contour is equal to
–ve of the time rate of change of the total magnetic flux that goes through any
arbitrary surface that is bounded by the closed contour
Points to Note: This law says that time changing magnetic fields can also generate
electric fields
The positive directions for the surface normal vector
and of the contour are related by the right hand rule
Michael Faraday
(1791-1867)
B
7. 7
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Faraday’s Law – Differential Form
( )∫∫ ×∇=∫ adAsdA
rrr
..
Stokes Theorem:
For any vector field:
The line integral of a vector over a closed contour is
equal to the surface integral of the curl of that vector
over any arbitrary surface that is bounded by the
closed contour
Using the Stokes Theorem with Farady’s Law in Integral Form:
( )
t
H
E
t
B
E
adB
t
adE
adB
t
sdE
o
∂
∂
−=×∇
∂
∂
−=×∇⇒
∫∫
∂
∂
−=∫∫ ×∇⇒
∫∫
∂
∂
−=∫
r
r
r
r
rrrr
rrrr
µ
or
..
..
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Ampere’s Law – Integral Form
∫∫
∂
∂
+∫∫=∫∫∫
∂
∂
+∫∫=∫ adD
t
adJsdHadE
t
adJsdH o
rrrrrrrrrrrr
...or... ε
What is this law saying ……??
Ampere’s Law: The line integral of magnetic field over a closed contour is equal
to the total current plus the time rate of change of the total electric flux that goes
through any arbitrary surface that is bounded by the closed contour
Points to Note: This law says that electrical currents and time
changing electric fields can generate magnetic fields. Since there
are no magnetic charges, this is the only known way to generate
magnetic fields
The positive directions for the surface normal vector
and of the contour are related by the right hand rule
electric flux density
electric current
density
A. M. Ampere
(1775-1836)
DJ
8. 8
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Ampere’s Law – Differential Form
( )∫∫ ×∇=∫ adAsdA
rrr
..
Stokes Theorem:
For any vector field:
The line integral of a vector over a closed contour is
equal to the surface integral of the curl of that vector
over any arbitrary surface that is bounded by the
closed contour
Using the Stokes Theorem with Ampere’s Law in Integral Form:
( )
t
E
JH
t
D
JH
adD
t
adJadH
adD
t
adJsdH
o
∂
∂
+=×∇
∂
∂
+=×∇⇒
∫∫
∂
∂
+∫∫=∫∫ ×∇⇒
∫∫
∂
∂
+∫∫=∫
r
rr
r
rr
rrrrrr
rrrrrr
ε
or
...
...
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Maxwell’s Equations and Light – Coupling of E and H Fields
0.
.
=∇
=∇
H
E
o
o
r
r
µ
ρε
t
E
JH
t
H
E
o
o
∂
∂
+=×∇
∂
∂
−=×∇
r
rr
r
r
ε
µ
Time varying electric and magnetic fields are coupled - this coupling is
responsible for the propagation of electromagnetic waves
Electromagnetic Wave Equation in Free Space:
Assume: and take the curl of the Faraday’s Law on both sides:0== J
r
ρ
( ) 2
2
t
E
t
H
t
H
E oo
oo
∂
∂
−=
∂
×∇∂
−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
×−∇=×∇×∇
rrr
r
εµ
µµ
( ) 2
2
t
E
E oo
∂
∂
−=×∇×∇⇒
r
r
εµ
( ) 2
2
2
1
t
E
c
E
∂
∂
−=×∇×∇⇒
r
r Equation for a wave traveling
at the speed c
m/s103
1 8
×≈=
oo
c
µε
9. 9
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Maxwell’s Equations and Light
( ) 2
2
2
1
t
E
c
E
∂
∂
−=×∇×∇
r
r
Equation for a wave traveling
at the speed c:
m/s103
1 8
×≈=
oo
c
µε
In 1865 Maxwell wrote:
“This velocity is so nearly that of light, that it seems we have strong
reason to conclude that light itself is an electromagnetic disturbance in
the form of waves propagated through the electromagnetic field
according to electromagnetic laws”
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Electrostatics and Magnetostatics
Suppose we restrict ourselves to time-independent situations (i.e. nothing is
varying with time – everything is stationary)
We get two sets of equations for electric and magnetic fields that are completely
independent and uncoupled:
( ) ( )rrEo
rrr
ρε =∇.
( ) 0=×∇ rE
rr
( ) 0. =∇ rHo
rr
µ
( ) JrH
rrr
=×∇
Equations of Electrostatics Equations of Magnetostatics
• Electric fields are produced by
only electric charges
• In electrostatics problems one
needs to determine electric field
given some charge distribution
• Magnetic fields are produced by
only electric currents
• In magnetostatics problems
one needs to determine magnetic
field given some current
distribution
10. 10
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Electroquasistatics and Magnetoquasistatics - I
• The restriction to completely time-independent situations is too limiting and often
unnecessary
• What if things are changing in time but “slowly” ……..(how slowly is “slowly” ?)
• Allowing for slow time variations, one often uses the equations of
electroquasistatics and magnetoquasistatics
( ) ( )trtrEo ,,.
rrr
ρε =∇
( ) 0, =×∇ trE
rr
( ) 0,. =∇ trHo
rr
µ
( ) ( )trJtrH ,,
rrrr
=×∇
Equations of Electroquasistatics Equations of Magnetoquasistatics
• Electric fields are produced by
only electric charges
• Once the electric field is
determined, the magnetic field can
be found by the last equation
• Magnetic fields are produced by
only electric currents
• Once the magnetic field is
determined, the electric field can be
found by the last equation
t
E
JH o
∂
∂
+=×∇
r
rr ε
t
H
E o
∂
∂
−=×∇
r
r µ
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Electroquasistatics and Magnetoquasistatics - II
Question from the last slide: How slowly is “slowly” ?
• Electromagnetic waves and signals move at the speed c (speed of light)
Answer: Time variations are considered slow if the time scales over which things
are changing are much longer compared to the time taken by light to cover
distances equal to the length scales of the problem
An amplifier chip operating
from 100 MHz to 10 GHz
3 cm
Example:
• Time scale of the problem = 1/(100 MHz) = 10 ns
• Length scale of the problem = 3 cm
• Time taken by light to travel 3 cm = 0.1 ns
Since 10 ns >> 0.1 ns, quasistatics is a valid
means of analysis at 100 MHz
• Time scale of the problem = 1/(10 GHz) = 0.1 ns
• Length scale of the problem = 3 cm
• Time taken by light to travel 3 cm = 0.1 ns
Quasistatics is not a valid means of analysis at
10 GHz
100 MHz Operation
10 GHz Operation
11. 11
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Electroquasistatics and Magnetoquasistatics - III
Question (contd..): How slowly is “slowly” ?
Electromagnetic wave frequency f and wavelength λ are related to the speed
of the wave c by the relation:
cf =λ
Let: L = length scale of the problem
T = time scale of the problem ≈ 1/f
Condition for quasitatic analysis to be valid:
L
L
f
c
LcT
c
L
T
>>⇒
>>⇒
>>⇒
>>
λ
Quasistatic analysis is valid if the wavelength of electromagnetic wave at the
frequency of interest is much longer than the length scales involved in the problem
ECE 303 – Fall 2007 – Farhan Rana – Cornell University