2/12/07184 Lecture 191 PHY 184 Spring 2007 Lecture 19 Title: Kirchoff’s Rules for Circuits.
Spring 2016 Lecture 12 - eee.iub.edu.bd
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IUB Dr. Mustafa H. Chowdhury 1
HF Electromagnetic Waves
Lecture – 12
Spring 2016
Instructor
Dr. Mustafa H. Chowdhury
Curl, Magnetic Flux
&
Maxwell’s Equations
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Curl
Now, our objective is to obtain the point form of Ampere’s
circuital law.
An incremental closed path in Cartesian coordinates is selected
for the application of Ampere’s circuital law in order determine
the spatial rate of change of H as shown in figure.
zzyyxx HHH aaaH 0000
We assume that some current, as
yet unspecified, produces a
reference value for H at the
center of this small rectangle.
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Curl (contd..)
The closed line integral of H about this path is then
approximately the sum of the four values of H • L on each side.
We choose the direction of traverse as 1-2-3-4-1, which
corresponds to a current in the a direction az, and the first
contribution therefore
where Hy,1-2 can be written in terms
of Hy as
yH y 21,21LH
x
x
HHH
y
yy2
1021,
(1)
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Curl (contd..)
Therefore, equation (1) can be written as
Along the next section of the path we have
xyy
HHxH x
xx
2
1032,32 LH
yxx
HH
y
y
2
1021 LH (2)
(3)
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Curl (contd..)
Similarly, for the remaining two segments
Adding, (2), (3), (4), (5), we have
yxx
HH
y
y
2
1043 LH
xyy
HH x
x
2
1014 LH
(4)
(5)
yxy
H
x
Hd xy
LH
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Curl (contd..)
By Ampere’s circuital law, this result must be equal to the current
enclosed by the path, or the current crossing any surface bounded
by the path.
If we assume a general current density Jz, the enclosed current is
then I = Jz x y, and therefore
or
yxJyxy
H
x
Hd z
xy
LH
zxy
Jyxy
H
x
H
yx
d
LH
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Curl (contd..)
If the closed path shrink path tends to zero, the above equation
will be more exact. By taking the limit,
If we choose closed paths which are oriented perpendicularly to
each of the remaining two coordinate axes, analogous processes
lead to expressions for the y and z components of the current
density,
zxy
yxJ
y
H
x
H
yx
d
LH
0,lim
x
yz
zyJ
z
H
y
H
zy
d
LH
0,lim
(6)
(7)
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Curl (contd..)
And
Comparing (6), (7), and (8), we see that a component of the
current density is given by the limit of the quotient of the closed
line integral of H about a small path in a plane normal to that
component and of the area enclosed as the path shrinks to zero.
This limit has its counterpart in other fields of science and long
ago received the name of curl.
The curl of any vector is a vector, and any component of the curl
is given by the limit of the quotient of the closed line integral of
the vector about a small path in a plane normal to that component
desired and the area enclosed, as the path shrinks to zero.
yzx
xzJ
x
H
z
H
xz
d
LH
0,lim (8)
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Curl (contd..)
It should be noted that the above definition of curl does not refer
specifically to a particular coordinate system.
The mathematical form of the definition is
where SN is the planar area enclosed by the closed line integral.
The subscript N indicates that the component of the curl is that
component which is normal to the surface enclosed by the closed
path. It may represent any component in any coordinate system.
N
SN
S
d
N
LHH
0limcurl (9)
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Curl (contd..)
In Cartesian coordinates the definition (9) shows that the x, y, and
z components of the curl of H are given by (6), (7), and (8), and
therefore
This result may be written in the form of a determinant,
zxy
yzx
x
yz
y
H
x
H
x
H
z
H
z
H
y
HaaaHH
curl
zyx
zyx
HHH
zyx
aaa
HHcurl
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Curl (contd..)
Therefore, the curl in different coordinate systems are
Cartesiany
H
x
H
x
H
z
H
z
H
y
Hz
xy
yzx
x
yz aaaH
lCylindrica
y
HHH
z
H
z
HHz
xzz aaaH
11
Spherical
H
r
rH
r
r
rHH
rz
HHH
r
r
rr
z
a
aaH
1
sin
11sin
sin
1
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Curl (contd..)
Now in Cartesian coordinate system
Therefore, combining (6), (7), (8) and (10)
So
This is the point form of Ampere’s circuital law and the 2nd
Maxwell’s equation.
zxy
yzx
x
yz
y
H
x
H
x
H
z
H
z
H
y
HaaaH
(10)
zyx JJJ H
JH
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Maxwell’s 2nd equation
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Curl (contd..)
According to Kirchhoff’s voltage law
Using Stroke’s theorem
or
Finally,
0 LE d
0 E Maxwell’s 3rd equation
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SELE ddS
0SE dS
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Magnetic flux density
In free space, the magnetic flux density is defined as
where 0 is the permeability of free space and is given by
The unit of B in mks unit is Wb/m2 or in SI unit is Tesla (T).
HB 0
H/m104 7
0
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Magnetic flux
Magnetic flux is defined as the flux passing through any
designated area,
Our analogy should now remind us of the electric flux E,
measured coulombs, and of Gauss's law, which states that the
total flux passing through any closed surface is equal to the
charge enclosed,
The charge Q is the source of the lines of electric flux and these
lines begin and terminate on positive and negative charge,
respectively.
S
dSB
QdS
SD
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Magnetic flux (contd..)
But, no such source has ever been discovered for the lines of
magnetic flux. In the example of the infinitely long straight
filament carrying a direct current I, the H field formed concentric
circles about the filament. Since B = µ0H, the B field is of the
same form. The magnetic flux lines are closed and do not
terminate on “magnetic charge”. For this reason Gauss's, law for
the magnetic field is
Using the divergence theorem
0S dSB
volS
dvdS BB
0 B Maxwell’s 4th equation
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Maxwell’s equations
The point form of Maxwell’s equations for static electric and
magnetic fields are
And the corresponding 4 sets of Maxwell’s equations in integral
form
0
0
B
JH
E
D v
SJLH
SB
LE
SD
dId
d
d
dvQd
S
S
vvolS
0
0
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Example 1
Find the magnetic flux contained between the conductors in a
length d is the flux crossing any radial plane extending from =
a to = b and from z = 0 to z = d.
Solution :
aHB2
00
I
a
bId
dzdI
db
a
d
S
ln2
2
0
0
0
aaSB
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Assignment-10
A solid nonmagnetic conductor of circular cross section has its
axis on the z axis and carries a uniformly distributed total current
of 60 A in the z direction. If the radius is 4 mm, find: (a) B at =
5 mm; (b) B at = 3 mm; (c) the total magnetic flux per unit
length inside the conductor; (d) the total flux per unit length
outside the conductor.
Ans : 2.40 mT; 2.25 mT; 6 µWb;
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Stroke’s theorem
Consider the surface S as shown in Fig. Below, which is broken
up into incremental surface of area S. The sum of the closed
line integrals about the perimeter of every S is the same as the
closed line integral about the perimeter of S.
If we apply the
definition of the curl to
one of these incremental
surfaces, then
NN
S
S
daHH
LH
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Therefore,
where aN is a unit vector in the direction of the right-hand normal
to S.
Finally, we have
This equation is known as the Stroke’s theorem.
Stroke’s theorem (contd..)
SHaHLH Sd NS
SHLH ddS
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An alternating way of deriving the Stroke’s theorem is
Therefore,
Stroke’s theorem (contd..)
SH
SJ
LH
d
d
Id
S
S
SHLH ddS
SJ dIS
JH
since
since
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Next class
Next Class
On
Magnetic
Materials
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