Download Magnetic Flux Density (Cont`d)

EEE 498/598
Overview of Electrical
Engineering
Lecture 7: Magnetostatics: Ampere’s Law
Of Force; Magnetic Flux Density; Lorentz
Force; Biot-savart Law; Applications Of
Ampere’s Law In Integral Form; Vector
Magnetic Potential; Magnetic Dipole;
Magnetic Flux
1
Lecture 7 Objectives

To begin our study of magnetostatics with
Ampere’s law of force; magnetic flux
density; Lorentz force; Biot-Savart law;
applications of Ampere’s law in integral
form; vector magnetic potential; magnetic
dipole; and magnetic flux.
2
Lecture 7
Overview of Electromagnetics
Fundamental laws of
classical electromagnetics
Special
cases
Electrostatics
Statics:
Input from
other
disciplines
Maxwell’s
equations
Magnetostatics
Electromagnetic
waves

0
t
Geometric
Optics
Transmission
Line
Theory
Circuit
Theory
Kirchoff’s
Laws
3
d  
Lecture 7
Magnetostatics



Magnetostatics is the branch of electromagnetics
dealing with the effects of electric charges in steady
motion (i.e, steady current or DC).
The fundamental law of magnetostatics is
Ampere’s law of force.
Ampere’s law of force is analogous to Coulomb’s
law in electrostatics.
4
Lecture 7
Magnetostatics (Cont’d)

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
5
Lecture 7
Ampere’s Law of Force



Ampere’s law of force is the “law of action”
between current carrying circuits.
Ampere’s law of force gives the magnetic force
between two current carrying circuits in an
otherwise empty universe.
Ampere’s law of force involves complete circuits
since current must flow in closed loops.
6
Lecture 7
Ampere’s Law of Force (Cont’d)

Experimental facts:
F21 F12
Two parallel wires
carrying current in the
same direction attract.
 Two parallel wires
carrying current in the
opposite directions
repel.


7
I1
F21

I2
F12

I1

I2
Lecture 7
Ampere’s Law of Force (Cont’d)

Experimental facts:

A short currentcarrying wire oriented
perpendicular to a
long current-carrying
wire experiences no
force.
8
F12 = 0

I2
I1
Lecture 7
Ampere’s Law of Force (Cont’d)

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.
9
Lecture 7
Ampere’s Law of Force (Cont’d)

The direction of the force established by the
experimental facts can be mathematically
represented by
unit vector in direction
of current I2

unit vector in direction
of current I1
aˆ F12  aˆ 2  aˆ1  aˆ R12
unit vector in
direction of force on
I2 due to I1

unit vector in direction
of I2 from I1
10
Lecture 7
Ampere’s Law of Force (Cont’d)

The force acting on a current element I2 dl2 by a
current element I1 dl1 is given by
0 I 2 d l 2  I1d l 1  aˆ R
F 12 
2
4
R12
12

Permeability of free space
0 = 4  10-7 F/m
11
Lecture 7
Ampere’s Law of Force (Cont’d)

The total force acting on a circuit C2 having a
current I2 by a circuit C1 having current I1 is
given by
d l 2  d l 1  aˆ R
 0 I1 I 2
F 12 
2


4 C C
R12
12
2

1
12
Lecture 7
Ampere’s Law of Force (Cont’d)

The force on C1 due to C2 is equal in
magnitude but opposite in direction to the
force on C2 due to C1.
F 21   F 12
13
Lecture 7
Magnetic Flux Density



Ampere’s force law describes an “action at a
distance” analogous to Coulomb’s law.
In Coulomb’s law, it was useful to introduce the
concept of an electric field to describe the
interaction between the charges.
In Ampere’s law, we can define an appropriate
field that may be regarded as the means by
which currents exert force on each other.
14
Lecture 7
Magnetic Flux Density (Cont’d)

The magnetic flux density can be introduced
by writing
0
F 12   I 2 d l 2 
4
C
2

C1
I d l
1
1
 aˆ R12

2
12
R
  I 2 d l 2  B12
C2
15
Lecture 7
Magnetic Flux Density (Cont’d)

where
0
B12 
4

I1d l 1  aˆ R12
C1
2
12
R
the magnetic flux density at the location of
dl2 due to the current I1 in C1
16
Lecture 7
Magnetic Flux Density (Cont’d)

Suppose that an infinitesimal current element Idl
is immersed in a region of magnetic flux density
B. The current element experiences a force dF
given by
d F  Id l  B
17
Lecture 7
Magnetic Flux Density (Cont’d)

The total force exerted on a circuit C carrying
current I that is immersed in a magnetic flux
density B is given by
F  I  dl  B
C
18
Lecture 7
Force on a Moving Charge

A moving point charge placed in a magnetic
field experiences a force given by
F m  Qv  B
Q
v
Id l  Qv
B
19
The force experienced
by the point charge is
in the direction into the
paper.
Lecture 7
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
F  F e  F m  qE  v  B

The Lorentz force equation is useful for
determining the equation of motion for electrons in
electromagnetic deflection systems such as CRTs.
20
Lecture 7
The Biot-Savart Law
The Biot-Savart law gives us the B-field
arising at a specified point P from a given
current distribution.
 It is a fundamental law of magnetostatics.

21
Lecture 7
The Biot-Savart Law (Cont’d)

The contribution to the B-field at a point P
from a differential current element Idl’ is given
by
0 I d l   R
d B(r ) 
3
4 R
22
Lecture 7
The Biot-Savart Law (Cont’d)
P
R
Id l 
r
r
23
Lecture 7
The Biot-Savart Law (Cont’d)

The total magnetic flux at the point P due to the
entire circuit C is given by
0 I d l   R
B(r )  
3
4 R
C
24
Lecture 7
Types of Current Distributions



Line 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.
25
Lecture 7
The Biot-Savart Law (Cont’d)

For a surface distribution of current, the B-S law
becomes
0 J s r  R
B(r )  
ds
3
4 R
S

For a volume distribution of current, the B-S law
becomes
0 J r  R
B(r )  
dv
3
4 R
V
26
Lecture 7
Ampere’s Circuital Law in
Integral Form

Ampere’s Circuital Law in integral form
states that “the circulation of the magnetic
flux density in free space is proportional to
the total current through the surface
bounding the path over which the circulation
is computed.”
B

d
l


I
0
encl

C
27
Lecture 7
Ampere’s Circuital Law in
Integral Form (Cont’d)
By convention, dS is
taken to be in the
direction defined by the
right-hand rule applied
to dl.
dl
dS
S
I encl   J  d s
S
Since volume current
density is the most
general, we can write
Iencl in this way.
28
Lecture 7
Ampere’s Law and Gauss’s Law


Just as Gauss’s law follows from Coulomb’s law,
so Ampere’s circuital law follows from Ampere’s
force law.
Just as Gauss’s law can be used to derive the
electrostatic field from symmetric charge
distributions, so Ampere’s law can be used to
derive the magnetostatic field from symmetric
current distributions.
29
Lecture 7
Applications of Ampere’s Law

Ampere’s law in integral form is an integral
equation for the unknown magnetic flux density
resulting from a given current distribution.
B

d
l


I
0
encl

C
known
unknown
30
Lecture 7
Applications of Ampere’s Law
(Cont’d)
In general, solutions to integral equations
must be obtained using numerical
techniques.
 However, for certain symmetric current
distributions closed form solutions to
Ampere’s law can be obtained.

31
Lecture 7
Applications of Ampere’s Law
(Cont’d)
Closed form solution to Ampere’s law
relies on our ability to construct a suitable
family of Amperian paths.
 An Amperian path is a closed contour to
which the magnetic flux density is
tangential and over which equal to a
constant value.

32
Lecture 7
Magnetic Flux Density of an Infinite
Line Current Using Ampere’s Law
Consider an infinite line current along the z-axis
carrying current in the +z-direction:
I
33
Lecture 7
Magnetic Flux Density of an Infinite Line
Current Using Ampere’s Law (Cont’d)
(1) Assume from symmetry and the right-hand
rule the form of the field
B  aˆ B  
(2) Construct a family of Amperian paths
circles of radius  where
34
Lecture 7
Magnetic Flux Density of an Infinite Line
Current Using Ampere’s Law (Cont’d)
(3) Evaluate the total current passing through the
surface bounded by the Amperian path
I encl   J  d s
S
35
Lecture 7
Magnetic Flux Density of an Infinite Line
Current Using Ampere’s Law (Cont’d)
y
Amperian path

x
I
I encl  I
36
Lecture 7
Magnetic Flux Density of an Infinite Line
Current Using Ampere’s Law (Cont’d)
(4) For each Amperian path, evaluate the integral
 B  d l  Bl
C
magnitude of B
on Amperian
path.
length
of Amperian
path.
 B  d l  B   2 
C
37
Lecture 7
Magnetic Flux Density of an Infinite Line
Current Using Ampere’s Law (Cont’d)
(5) Solve for B on each Amperian path
B
0 I encl
l
0 I
B  aˆ
2 
38
Lecture 7
Applying Stokes’s Theorem to
Ampere’s Law
 B  dl    B d s
C
S
 0 I encl  0  J  d s
S
 Because the above must hold for any
surface S, we must have
Differential form
of Ampere’s Law
  B  0 J
39
Lecture 7
Ampere’s Law in Differential
Form

Ampere’s law in differential form implies
that the B-field is conservative outside of
regions where current is flowing.
40
Lecture 7
Fundamental Postulates of
Magnetostatics

Ampere’s law in differential form
  B  0 J

No isolated magnetic charges
 B  0
41
B is solenoidal
Lecture 7
Vector Magnetic Potential

Vector identity: “the divergence of the curl of
any vector field is identically zero.”
    A  0

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.”
42
Lecture 7
Vector Magnetic Potential
(Cont’d)

Since the magnetic flux density is
solenoidal, it can be written as the curl of
a vector field called the vector magnetic
potential.
 B  0
 B   A
43
Lecture 7
Vector Magnetic Potential
(Cont’d)

The general form of the B-S law is
0 J r  R
B(r )  
dv
3
4 R
V

Note that
R
1
    3
R
R
44
Lecture 7
Vector Magnetic Potential
(Cont’d)

Furthermore, note that the del operator operates
only on the unprimed coordinates so that
J r  R
1

  J r   
3
R
R
1
    J r 
R
 J r  
 

 R 
45
Lecture 7
Vector Magnetic Potential
(Cont’d)

Hence, we have
0 J r 

Br    
d
v

4 V  R
Ar 
46
Lecture 7
Vector Magnetic Potential
(Cont’d)

For a surface distribution of current, the vector
magnetic potential is given by
0
A(r ) 
4

J s r 

d
s
S  R
For a line current, the vector magnetic potential is
given by
0 I d l 
A(r ) 

4 L R
47
Lecture 7
Vector Magnetic Potential
(Cont’d)
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 find B.
 In some ways, the vector magnetic
potential A is analogous to the scalar
electric potential V.

48
Lecture 7
Vector Magnetic Potential
(Cont’d)
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.

49
Lecture 7
Magnetic Dipole


A magnetic dipole comprises 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 as
classical physics is concerned).
50
Lecture 7
Magnetic Dipole (Cont’d)
The magnetic dipole is analogous to the
electric dipole.
 Just as the electric dipole is useful in
helping us to understand the behavior of
dielectric materials, so the magnetic dipole
is useful in helping us to understand the
behavior of magnetic materials.

51
Lecture 7
Magnetic Dipole (Cont’d)

Consider a small circular loop of radius b
carrying a steady current I. Assume that the
wire radius has a negligible cross-section.
y
x
b
52
Lecture 7
Magnetic Dipole (Cont’d)

The vector magnetic potential is evaluated
for R >> b as
0 I 2 aˆ bd 
A(r ) 
R
4 0
 0 Ib 2
1 b sin  cos   


d 
 aˆ x sin    aˆ y cos   

2


r
4 0

r
 0 Ib
b sin 

 aˆ x sin   aˆ y cos  

r2
4
0 Ib 2
sin 
 aˆ
2
4 r
53
Lecture 7
Magnetic Dipole (Cont’d)

The magnetic flux density is evaluated for
R >> b as
0
2
ˆ
ˆ



B   A 
I

b
a
2
cos


a
r
 sin  
3
4 r
54
Lecture 7
Magnetic Dipole (Cont’d)

Recall electric dipole
p
aˆr 2 cos   aˆ sin  
E
3
40 r
p  electric dipole moment  Qd

The electric field due to the electric charge
dipole and the magnetic field due to the
magnetic dipole are dual quantities.
55
Lecture 7
Magnetic Dipole Moment

The magnetic dipole moment can be defined
as
2
m  aˆ z Ib
Direction of the dipole moment
is determined by the direction
of current using the right-hand
rule.
56
Magnitude of
the dipole
moment is the
product of the
current and
the area of the
loop.
Lecture 7
Magnetic Dipole Moment
(Cont’d)

We can write the vector magnetic potential in
terms of the magnetic dipole moment as
0 m sin  0 m  aˆ r
A  aˆ

2
2
4 r
4 r
We can write the B field in terms of the
magnetic dipole moment as
0
0 
 1 
B
maˆr 2 cos   aˆ sin   
 m   
3
4 r
4 
 r 

57
Lecture 7
Divergence of B-Field

The B-field is solenoidal, i.e. the
divergence of the B-field is identically
equal to zero:
 B  0
Physically, this means that magnetic
charges (monopoles) do not exist.
 A magnetic charge can be viewed as an
isolated magnetic pole.

58
Lecture 7
Divergence of B-Field (Cont’d)


No matter how small
the magnetic is
divided, it always has
a north pole and a
south pole.
The elementary
source of magnetic
field is a magnetic
dipole.
N
N
S
S
N
S
N
I
59
S
Lecture 7
Magnetic Flux

The magnetic flux
crossing an open
surface S is given by
B
   Bds
Wb
S
S
C
Wb/m2
60
Lecture 7
Magnetic Flux (Cont’d)

From the divergence theorem, we have
  B  0     B dv  0   B  d s  0
V

S
Hence, the net magnetic flux leaving any
closed surface is zero. This is another
manifestation of the fact that there are no
magnetic charges.
61
Lecture 7
Magnetic Flux and Vector
Magnetic Potential

The magnetic flux across an open surface
may be evaluated in terms of the vector
magnetic potential using Stokes’s theorem:
   B  d s    A d s
S
S
  A dl
C
62
Lecture 7