lect07 emt
TRANSCRIPT
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1
EEE 498/598
Overview of Electrical Engineering
Lecture 7: Magnetostatics: Amperes LawOf Force; Magnetic Flux Density; Lorentz
Force; Biot-savart Law; Applications OfAmperes Law In Integral Form; Vector
Magnetic Potential; Magnetic Dipole;Magnetic Flux
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Lecture 72
Lecture 7 Objectives
To begin our study of magnetostatics with
Amperes law of force; magnetic flux
density; Lorentz force; Biot-Savart law;applications of Amperes law in integral
form; vector magnetic potential; magnetic
dipole; and magnetic flux.
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Lecture 73
Overview of Electromagnetics
Maxwells
equations
Fundamental laws of
classical electromagnetics
Special
cases
Electro-
statics
Magneto-
staticsElectro-
magnetic
waves
Kirchoffs
Laws
Statics: 0
t
d
Geometric
Optics
Transmission
Line
Theory
CircuitTheory
Input fromother
disciplines
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Lecture 74
Magnetostatics
Magnetostaticsis the branch of electromagnetics
dealing with the effects of electric charges in steady
motion (i.e, steady current or DC).
The fundamental law of magnetostaticsis
Amperes law of force.
Amperes law of forceis analogous to Coulombs
lawin electrostatics.
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Lecture 75
Magnetostatics (Contd)
In magnetostatics, the magnetic field is
produced by steady currents. The
magnetostatic field does not allow for inductive coupling between circuits
coupling between electric and magnetic fields
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Lecture 76
Amperes Law of Force
Amperes law of forceis the law of action
between current carrying circuits.
Amperes law of forcegives the magnetic force
between two current carrying circuitsin an
otherwise empty universe.
Amperes law of force involves complete circuits
since current must flow in closed loops.
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Lecture 77
Amperes Law of Force (Contd)
Experimental facts:
Two parallel wires
carrying current in thesame direction attract.
Two parallel wires
carrying current in the
opposite directionsrepel.
I1 I2
F12F21
I1 I2
F12F21
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Lecture 78
Amperes Law of Force (Contd)
Experimental facts:
A short current-
carrying wire orientedperpendicular to a
long current-carrying
wire experiences no
force.
I1
F12 = 0
I2
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Lecture 79
Amperes Law of Force (Contd)
Experimental facts:
The magnitude of the force is inversely
proportional to the distance squared.The magnitude of the force is proportional to
the product of the currents carried by the two
wires.
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Lecture 710
Amperes Law of Force (Contd)
The direction of the force established by the
experimental facts can be mathematically
represented by
1212
12 RF aaaa
unit vector in
direction of force on
I2due toI1
unit vector in direction
ofI2fromI1
unit vector in directionof currentI1
unit vector in directionof currentI2
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Lecture 711
Amperes Law of Force (Contd)
The force acting on a current elementI2 dl2by a
current elementI1 dl1is given by
2
12
1122012
12
4 R
aldIldIF R
Permeability of free space
0= 410-7 F/m
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Lecture 712
Amperes Law of Force (Contd)
The total force acting on a circuit C2having a
current I2by a circuit C1having current I1is
given by
2 1
12
2
12
1221012
4C C
R
R
aldldIIF
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Lecture 713
Amperes Law of Force (Contd)
The force on C1due to C2is equal in
magnitude but opposite in direction to the
force on C2due to C1.
1221 FF
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Lecture 714
Magnetic Flux Density
Amperes force law describes an action at a
distance analogous to Coulombs law.
In Coulombs law, it was useful to introduce the
concept of an electric fieldto describe the
interaction between the charges.
In Amperes law, we can define an appropriate
field that may be regarded as the means by
which currents exert force on each other.
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Lecture 715
Magnetic Flux Density (Contd)
The magnetic f lux densitycan be introduced
by writing
2
2 1
12
1222
2
12
1102212
4
C
C C
R
BldI
R
aldIldIF
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Lecture 716
Magnetic Flux Density (Contd)
where
1
1 22
12
11012
4C
R
R
aldIB
the magnetic flux density at the location ofdl2due to the currentI1in C1
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Lecture 717
Magnetic Flux Density (Contd)
Suppose that an infinitesimal current elementIdl
is immersed in a region of magnetic flux density
B. The current element experiences a force dF
given by
BlIdFd
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Lecture 718
Magnetic Flux Density (Contd)
The total force exerted on a circuit Ccarrying
current Ithat is immersed in a magnetic flux
density Bisgiven by
C
BldIF
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Lecture 719
Force on a Moving Charge
A moving point charge placed in a magnetic
field experiences a force given by
BvQ
The force experienced
by the point charge isin the direction into the
paper.
BvQFm vQlId
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Lecture 720
Lorentz Force
If a point charge is moving in a region where both
electric and magnetic fields exist, then it experiences
a total force given by
The Lorentz force equation is useful fordetermining the equation of motion for electrons in
electromagnetic deflection systems such as CRTs.
BvEqFFF me
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Lecture 721
The Biot-Savart Law
The Biot-Savart lawgives us theB-field
arising at a specified pointPfrom a given
current distribution. It is a fundamental law of magnetostatics.
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Lecture 722
The Biot-Savart Law (Contd)
The contribution to theB-field at a pointP
from a differential current elementIdlis given
by
3
0
4
)(
R
RldIrBd
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Lecture 723
The Biot-Savart Law (Contd)
lId
PR
r r
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Lecture 724
The Biot-Savart Law (Contd)
The total magnetic flux at the pointPdue to the
entire circuit Cis given by
C
R
RldIrB
3
0
4)(
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Lecture 725
Types of Current Distributions
L ine current density(current) - occurs for
infinitesimally thin filamentary bodies (i.e.,
wires of negligible diameter).
Surface current density(current per unit
width) - occurs when body is perfectly
conducting.
Volume current density(current per unit
cross sectional area) - most general.
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Lecture 726
The Biot-Savart Law (Contd) For a surface distribution of current, the B-S law
becomes
For a volume distribution of current, the B-S law
becomes
S
s sdR
RrJrB
3
0
4)(
V
vdR
RrJrB3
0
4)(
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Lecture 727
Amperes Circuital Law in
Integral Form
Amperes Circuital Lawin integral formstates that the circulation of the magneticflux density in free space is proportional to
the total current through the surfacebounding the path over which the circulationis computed.
encl
C
IldB 0
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Lecture 728
Amperes Circuital Law in
Integral Form (Contd)
By convention, dSis
taken to be in the
direction defined by the
right-hand rule appliedto dl.
S
encl sdJI
Since volume current
density is the mostgeneral, we can write
Ienclin this way.
S
dl
dS
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Lecture 729
Amperes Law and Gausss Law
Just as Gausss law follows from Coulombs law,
so Amperes circuital law follows from Amperes
force law.
Just as Gausss law can be used to derive the
electrostatic field from symmetric charge
distributions, so Amperes law can be used to
derive the magnetostatic field from symmetriccurrent distributions.
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Lecture 730
Applications of Amperes Law
Amperes law in integral form is an integral
equationfor the unknown magnetic flux density
resulting from a given current distribution.
encl
C
IldB 0
known
unknown
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Lecture 731
Applications of Amperes Law
(Contd)
In general, solutions to integral equations
must be obtained using numerical
techniques. However, for certain symmetric current
distributions closed form solutions to
Amperes law can be obtained.
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Lecture 732
Applications of Amperes Law
(Contd)
Closed form solution to Amperes law
relies on our ability to construct a suitable
family ofAmperian paths
.AnAmperian pathis a closed contour to
which the magnetic flux density is
tangential and over which equal to aconstant value.
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Lecture 733
Magnetic Flux Density of an Infinite
Line Current Using Amperes Law
Consider an infinite line current along the z-axis
carrying current in the +z-direction:
I
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Lecture 734
Magnetic Flux Density of an Infinite Line
Current Using Amperes Law (Contd)
(1) Assume from symmetry and the right-hand rule
the form of the field
(2) Construct a family of Amperian paths
BaB
circles of radius where
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Lecture 735
Magnetic Flux Density of an Infinite Line
Current Using Amperes Law (Contd)
(3) Evaluate the total current passing through the
surface bounded by the Amperian path
S
encl sdJI
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Lecture 736
Magnetic Flux Density of an Infinite Line
Current Using Amperes Law (Contd)
Amperian path
IIencl
I
x
y
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Lecture 737
Magnetic Flux Density of an Infinite Line
Current Using Amperes Law (Contd)
(4) For each Amperian path, evaluate the integral
BlldBC
2BldBC
magnitude ofB
on Amperian
path.
lengthof Amperian
path.
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Lecture 738
Magnetic Flux Density of an Infinite Line
Current Using Amperes Law (Contd)
(5) Solve for Bon each Amperian path
lIB encl0
2
0IaB
A l i S k Th
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Lecture 739
Applying Stokess Theorem to
Amperes Law
S
encl
SC
sdJI
sdBldB
00
Because the above must hold for any
surface S, we must have
JB 0Differential form
of Amperes Law
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Lecture 740
Amperes Law in Differential
Form
Amperes law in differential form implies
that theB-field is conservativeoutside of
regions where current is flowing.
d l l f
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Lecture 741
Fundamental Postulates of
MagnetostaticsAmperes law in differential form
No isolated magnetic charges
JB 0
0
B
Bis solenoidal
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Lecture 742
Vector Magnetic Potential
Vector identity: the divergence of the curl of
any vector field is identically zero.
Corollary: If the divergence of a vector field is
identically zero, then that vector field can be
written as the curl of some vector potential
field.
0 A
V M i P i l
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Lecture 743
Vector Magnetic Potential
(Contd)
Since the magnetic flux density is
solenoidal, it can be written as the curl of
a vector field called the vector magnetic
potential.
ABB 0
V M i P i l
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Lecture 744
Vector Magnetic Potential
(Contd)The general form of the B-S law is
Note that
V
vd
R
RrJrB
3
0
4
)(
31
RR
R
Vector Magnetic Potential
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Lecture 745
Vector Magnetic Potential
(Contd) Furthermore, note that the deloperator operates
only on the unprimed coordinates so that
R
rJ
rJR
RrJ
RRrJ
1
13
V M i P i l
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Lecture 746
Vector Magnetic Potential
(Contd)
Hence, we have
vdR
rJrBV
4
0
rA
V M i P i l
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Lecture 747
Vector Magnetic Potential
(Contd) For a surface distribution of current, the vector
magnetic potential is given by
For a line current, the vector magnetic potential is
given by
sdR
rJ
rAS
s
4)( 0
L R
ldIrA
4)( 0
V M i P i l
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Lecture 748
Vector Magnetic Potential
(Contd)
In some cases, it is easier to evaluate the
vector magnetic potential and then use
B= A, rather than to use the B-S law
to directly findB.
In some ways, the vector magnetic
potential Ais analogous to the scalarelectric potential V.
V M i P i l
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Lecture 749
Vector Magnetic Potential
(Contd)
In classical physics, the vector magnetic
potential is viewed as an auxiliary function
with no physical meaning.
However, there are phenomena in
quantum mechanics that suggest that the
vector magnetic potential is a real (i.e.,measurable) field.
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Lecture 750
Magnetic Dipole
A magnetic dipolecomprises a small current
carrying loop.
The point charge (charge monopole) is the
simplest source of electrostatic field. The
magnetic dipole is the simplest source of
magnetostatic field. There is no such thing as
a magnetic monopole (at least as far asclassical physics is concerned).
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Lecture 751
Magnetic Dipole (Contd)
The magnetic dipole is analogous to the
electric dipole.
Just as the electric dipole is useful inhelping us to understand the behavior of
dielectric materials, so the magnetic dipole
is useful in helping us to understand thebehavior of magnetic materials.
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Lecture 752
Magnetic Dipole (Contd)
Consider a small circular loop of radius bcarrying a steady currentI. Assume that the wireradius has a negligible cross-section.
x
y
b
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Lecture 753
Magnetic Dipole (Contd)The vector magnetic potential is evaluated
forR >> bas
sin4
sin
cos
sin
4
cossin1cossin
4
4)(
2
2
0
2
0
2
0 2
0
2
0
0
r
bIa
r
b
aa
Ib
dr
b
raa
Ib
R
bdaIrA
yx
yx
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Lecture 754
Magnetic Dipole (Contd)
The magnetic flux density is evaluated for
R >> bas
sincos2
4
2
3
0 aabIr
AB r
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Lecture 755
Magnetic Dipole (Contd)
Recall electric dipole
The electric field due to the electric charge
dipole and the magnetic field due to the
magnetic dipole are dualquantities.
sincos24 30
aar
pE r
Qdp momentdipoleelectric
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Lecture 756
Magnetic Dipole Moment
The magnetic dipole moment can be defined
as2
bIam z
Direction of the dipole moment
is determined by the directionof current using the right-hand
rule.
Magnitude of
the dipole
moment is the
product of the
current and
the area of the
loop.
M i Di l M
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Lecture 757
Magnetic Dipole Moment
(Contd)We can write the vector magnetic potential in
terms of the magnetic dipole moment as
We can write the B field in terms of the
magnetic dipole moment as
2020 4
4
sin
r
am
r
maA r
rmaam
rB r
1
4sincos2
4
0
3
0
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Lecture 758
Divergence of B-Field
The B-field is solenoidal, i.e. thedivergence of the B-field is identically equalto zero:
Physically, this means that magneticcharges (monopoles) do not exist.
A magnetic charge can be viewed as anisolated magnetic pole.
0 B
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Lecture 759
Divergence of B-Field (Contd)
N
S
N
S
NS
No matter how small
the magnetic is
divided, it always has
a north pole and asouth pole.
The elementary
source of magneticfield is a magnetic
dipole.
I
N
S
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Lecture 760
Magnetic Flux
The magnetic flux
crossing an open
surface Sis given by
S
sdBS
B
C
Wb/m2Wb
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Lecture 761
Magnetic Flux (Contd)
From the divergence theorem, we have
Hence, the net magnetic flux leaving anyclosed surface is zero. This is another
manifestation of the fact that there are nomagnetic charges.
000
SV
sdBdvBB
M i Fl d V
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Magnetic Flux and Vector
Magnetic Potential
The magnetic flux across an open surface
may be evaluated in terms of the vector
magnetic potential using Stokess theorem:
C
SS
ldA
sdAsdB