Section 4.4 - Gausss Law
r
r
Recall: Electric Flux Density
D = E [C/m 2 ]
r
r
Definition: Electric flux () =
D dS [C]
Surface
an
Properties:
D
Electric flux , is directly proportional
to the number of field lines passing through an area
Electric flux beg
r
Magnetic Vector Potential A (5.4)
We can use the magnetic vector potential to calculate the magnetic field
(a 3rd approach complementing Amperes and Biot-Savarat Laws.
r
Electrostatics ( E
Definition
r
= 0)
Magnetostatics ( B
= 0)
r
A = Ax i + Ay + Az k
Elec 251
Foundation of Electrostatics: Electric Charge
Matter is composed of
compact particles that carry
electric charge:
Electron & Proton
The charge on electron and proton are observed to
be equal and opposite.
The electron is said to have negative
Magnetostatics (Chapter 5)
Magnetostatics is the branch of electromagnetics dealing with
the effects of electric charges in steady motion
(i.e, steady current or DC where d/dt = 0).
In magnetostatics, the magnetic field is produced by steady
currents.
Constitutive Relations
In free space
Field Laws
Gauss Law
I
S
D = v
Z
D dS =
v dV = Q
D = 0 E
B = 0 H
V
0 = 8.854 1012 F/m
Magnetic flux density B
I
0 = 4 107 H/m
B=0
B dS = 0
In materials
S
Faradays Law
B
t
Z
I
d
B dS
E d =
dt S
C
Amp`ere-Maxwell
D
H=J
^
z
z
dz
(x,y,z)
^
y
^
x
y
x
1
^
z
^
r
r sin d
^
z
r d
dr
z
^
dz
r
z
d
d
^
d
y
y
x
d
^
x
d
Differential Volumes and Surfaces
dx
dy
Feb. 16, 2010
Coordinate Transformations
f =
x = cos ; y = sin
1 (A ) 1 A Az
+
+
z
A=
x = r sin cos ; y = r sin sin ;
ELEC 251 D. Davis
Lecture 23
The time dependent potential (retarded potential)
In the previous lecture we have discussed the time dependent contribution of electric flux
to Amperes law. We will now investigate the time dependence of the potential function
ELEC 251 D. Davis
Lecture 17
Magnetic Circuits
It is a common practice to design structures that can make use of magnetic flux to
perform various tasks. Magnetic circuits are structures where the magnetic flux is
generated from current sources and the mag
ELEC 251 D. Davis
Lecture 1
Electric Forces and Fields due to point charges
Coulomb developed an experimental law to relate the force applied on a charge due to the
presence of another charge. This law indicates that electrical charges can create force on
ELEC 251 D. Davis
Lecture 9
Dielectrics and permittivity
Relative permittivity: Dielectrics are insulators, materials that have no free charge so that
current (conduction current to be more precise) is nearly zero within the medium.
Dielectric materials c
ELEC 251 D. Davis
Lecture 12
More Laplace and Poisson Equations and Image Theory
It is possible to have a problem where the geometry suggests a two dimensional, (or three
dimensional), variation in the electric potential. The general solution is obtained
ELEC 251 D. Davis
Lecture 18
Multi-element Magnetic Circuits
In the previous lecture, the basic principles of magnetic circuits were introduced. In this
lecture we will explore the analysis of more complicated magnetic circuits.
Magnetic circuits are ofte
Poissons and Laplaces Equations
Section 4.5.5
Two approaches for finding E and V due to a
given charge distribution:
1st approach: given charge distribution, find E and V using
v ,
r
r
k v R dv
E ( R )=
R3
V=?
r
P r
& V ( at P) = E . d l
refernce
This
Department of Electrical and Computer Engineering
ELEC 251 - Fundamentals of Applied Electromagnetics
Project # 2: Application of Friis Transmission Formula
Project Report Due Date: Thursday April 2, 2015 (during class)
1 Introduction
One of the fundament
Department of Electrical and Computer Engineering
ELEC 251 - Fundamentals of Applied Electromagnetics
Project # 1:
Application of Vector Calculus in Cylindrical and Spherical Coordinates
Project Report Due Date: Tuesday January 27, 2015 (during class)
1 I
Maxwells Equations for Time-Varying Fields
Faradays Law: Electromagnetic Induction
(Chapter 6)
In the early 1830s Michael Faraday made the observation
that a changing current IP in one electric circuit can
cause a current to appear (induce a current IS )
Poissons and Laplaces Equations
Section 4.5.5
Two approaches for finding E and V due to a
given charge distribution:
1st approach: given charge distribution, find E and V using
v ,
r
r
k v R dv
E ( R )=
R3
V=?
r
P r
& V ( at P) = E . d l
refernce
This
Section 4.4 - Gausss Law
r
r
Recall: Electric Flux Density
D = E [C/m 2 ]
r
r
Definition: Electric flux (
) =
D dS [C]
Surface
an
Properties:
D
Electric flux , is directly proportional
to the number of field lines passing through an area
Electric flux
r
Magnetic Vector Potential A (5.4)
We can use the magnetic vector potential to calculate the magnetic field
(a 3rd approach complementing Amperes and Biot-Savarat Laws.
r
Electrostatics ( E
Definition
r
= 0)
Magnetostatics ( B
= 0)
r
A = Ax i + Ay j + Az
Intro
Welcome to Elec 251
Abdel R. Sebak
Intro
Room: EV 15.179
e-mail: [email protected]
Office Hours: Tu and Thu 10:00 am-12:00 noon,
or by appointment
1
Marking Scheme
One Scheme
Assignments
0
Quizzes
5
In-class Group Quizzes
5
Project
5
Midterm Tes
Magnetic Field of Toroid
Applying Amperes law over contour C:
Amperes law states that the line integral of
H around a closed contour C is equal to the
current traversing the surface bounded by
the contour.
-ve sign follows RHR
The magnetic field outside t
Maxwells Equations for Time-Varying Fields (Chapt 6)
Faradays Law: Electromagnetic Induction
In the early 1830s Michael Faraday made the observation
that a changing current IP in one electric circuit can
cause current to appear (induce a current IS ) in
Poissons and Laplaces Equations
Two approaches for finding E and V due to a given charge
distribution:
1st approach: given charge distribution, find E and V using
v ,
r
r
k v R dv
E ( R )=
R3
V=?
r
P r
& V ( at P) = E . d l
refernce
This method is val
Magnetostatics (Chapter 5)
Magnetostatics is the branch of electromagnetics dealing with
the effects of electric charges in steady motion
(i.e, steady current or DC where d/dt = 0).
In magnetostatics, the magnetic field is produced by steady
currents.
Gausss Law: Electric flux ()
Properties:
r r
= D.dS
Surface
S is surface
of the box
Electric flux , is directly proportional
to the number of field lines passing through an area
Electric flux begins on positive charges and terminates on
negative charges
Circuits vs Electromagnetics
Circuits
You have studied a number of devices
(resistors, capacitors, op amps, etc.)
and in order to make "connections"
between them, you have had to be certain
that they were solidly connected by
wire/solder/metal.
1
Set#1
E
Concordia University Department of Electrical and Computer Engineering
ELEC 251 Fundamentals of Applied Electromagnetics
Summer 2014
Instructor: Dr. D. Davis
Course Website: Accessed using MYCONCORDIA portal
Course Email: [email protected] (Use this email)
ELEC 251 D. Davis
Lecture 19
Multi-Source Magnetic Circuits
Magnetic circuits can have more than one coil or source connected to the same structure.
Applications such as power transformers typically have more than one coil and the
relationship between sou
ELEC 251 D. Davis
Lecture 11
Laplace and Poissson Equations
It is often the case that full knowledge of the behavior of the electric field or electric
potential is not available to the engineer. Often there is information only for some regions
in space. I
ELEC 251 D. Davis
Lecture 3
Electric Forces and Fields due to surface charge distributions
Surface Charges: In most practical circuits and systems, there will be metallic surfaces
(for example, microstrip lines for IC and high frequency circuits, wires fo
ELEC 251 D. Davis
Lecture 20
Inductance and Magnetic Energy
The basic principles of magnetic circuits and magnetic field analysis have been
introduced in the previous lecture sections. We will now focus on the relationships
governing magnetic stored energ
ELEC 251 D. Davis
Lecture 5
Multiple Charge Distributions
In this lecture we will expand on the previous discussion of Gauss law and investigate
the analysis of electric fields due to multiple charge distributions and how Gauss law
may be applied to very
ELEC 251 D. Davis
Lecture 16
Magnetic forces, materials and boundary conditions
We have previously discusses that movement of charge (either rotational or translational)
will produce a magnetic field. This is true on both the large scale (e.g. DC current
ELEC 251 D. Davis
Lecture 6
Electric potential
In this section we will investigate another approach to analyzing the effect charge has on
space and other charges. Up to this point we have been looking at electric field (a vector
field that represents the