Engineering Electromagnetics
Lecture 11
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
2 0 1 3
President University
Erwin Sitompul
EEM 11/1
Chapter 8
The Steady Magnetic Field
Curl
 In our study of Gauss’s law, we applied it to a differential
volume element which led to the “Concept of Divergence.”
 We now apply Ampere’s circuital law to the perimeter of a
differential surface element and discuss the third and last of the
special derivatives of vector analysis, the curl.
 Our immediate objective is to obtain the point form of Ampere’s
circuital law.
President University
Erwin Sitompul
EEM 11/2
Chapter 8
The Steady Magnetic Field
Curl
 Again, we choose rectangular
coordinate, and an incremental closed
path of sides Δx and Δy is selected.
 We assume that some current
produces a reference value for H at
the center of this small rectangle,
given by
H 0  H x 0a x  H y 0a y  H z 0a z
 The closed line integral of H about this path is then
approximately the sum of the four values of H·ΔL on each side.
 We choose the direction of traverse as 1-2-3-4-1, which
corresponds to a current in the az direction.
 The first contribution, from section 1-2, is therefore
(H  L)1 2  H y ,1 2 y
President University
Erwin Sitompul
EEM 11/3
Chapter 8
The Steady Magnetic Field
Curl
 The value of Hy on section 1-2 may
be given in terms of the reference
value Hy0 at the center of the
rectangle, the rate of change of Hy
with x, and the distance Δx/2 from the
center to the midpoint of side 1-2.
H y  1 
H y ,1 2 H y 0 
 x 
x  2 
 Thus,
(H  L)12


1 H y
x  y
 H y0 
2 x


 The next contribution, from section 2-3, is given as
(H  L)23
President University
H x,23 (x)
1 H x


  H x0 
y  x
2 y


Erwin Sitompul
EEM 11/4
Chapter 8
The Steady Magnetic Field
Curl
 Further, section 3-4 will give
(H  L)34


1 H y
x  (y)
 H y0 
2 x


 Finally, section 4-1 will give
(H  L)41
1 H x


 H x 0  2 y y  x


 Adding the results from all 4 sections, we obtain
 H y H x 
 H  dL  x  y  xy
President University
Erwin Sitompul
EEM 11/5
Chapter 8
The Steady Magnetic Field
Curl
 By Ampere’s circuital law, this closed path integration of this
magnetic field intensity H 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 J, the enclosed current
is then
I J z xy
and
 H  dL
 H y H x


y
 x

 xy

J z xy
 Finally,
 H  dL
H y
xy
President University
H x

x
y
Jz
Erwin Sitompul
EEM 11/6
Chapter 8
The Steady Magnetic Field
Curl
 As we let the closed path shrink, the approximation becomes
more nearly exact, and we have the current density in z
direction.
H  dL H y H x

lim


 Jz
x , y 0 xy
x
y
 If we choose closed paths which are oriented perpendicularly to
each of the remaining two coordinate axes, analogous
processes lead to expression for the x and y components of the
current density
lim
 H  dL  H
lim
 H  dL  H
y , z 0
z , x 0
yz
z x
President University
z
y

H y
z
 Jx
H z

 Jy
z
x
x
Erwin Sitompul
EEM 11/7
Chapter 8
The Steady Magnetic Field
Curl
 Comparing all equation in the previous slide, we can conclude
that “a component of a current density in a certain direction 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 as the
area enclosed by the path shrinks to zero.”
 This limit has its counterpart in other fields of science and
received the name of curl.
 The mathematical form of the curl is
(curl H ) N  lim
S N 0
 H  dL
S N
 ΔSN is the planar area enclosed by the closed line integral,
while N subscript indicated that the component of the curl is
normal to the surface enclosed by the closed path.
President University
Erwin Sitompul
EEM 11/8
Chapter 8
The Steady Magnetic Field
Curl
 In rectangular coordinates, the curl H is given by
 H z H y
curl H  

z
 y

 H x H z

 ax  
x
 z

 H y H x


ay  
y

 x

 az

Rectangular
 This result may be written in the form of a determinant or in
terms of the vector operator, as follows
ax

curl H 
x
Hx
President University
ay

y
Hy
az

z
Hz
curl H    H
Erwin Sitompul
EEM 11/9
Chapter 8
The Steady Magnetic Field
Curl
 1 H z H 
 H  H z
H  


 a  
z 

  
 z
1   (  H ) H  
 

 az
  
 

 a

Cylindrical
1   ( H sin  ) H 
H 


 ar
r sin  

 
1  1 H r  (rH ) 
 

 a
r  sin  
r 
1   (rH ) H r 
 

 a
r  r
 
Spherical
President University
EEM 11/10
Erwin Sitompul
Chapter 8
The Steady Magnetic Field
Small Paddle Wheel as a Curl Meter
×
• Clockwise rotation
• No rotation
H
I
2
a
1  (  H )
curl H  
a 
az  0
z
 
H
President University
Erwin Sitompul
EEM 11/11
Chapter 8
The Steady Magnetic Field
Curl
 Example
Let H = 0.2z2 ax for z>0, and H=0
elsewhere, as shown in the next
figure. Calculate  H·dL about a
square path with side d, centered
at (0,0,z1) in the y=0 plane where
z1 >d/2.
2
1
H

d
L

0.2(
z

d
)
d 0
1
2

 0.2( z1  12 d ) 2 d  0
 0.4z1d 2
(  H ) y
H  dL

 lim
ax
  H   x
d2
0.4 z1d 2
 lim
 0.4 z1
2
d 0
d
d 0
President University
Erwin Sitompul
ay
az
 y  z
0.2 z 2
0
0


2
 (0.2 z )a y  (0.2 z 2 )a z
z
y
 0.4 za y
EEM 11/12
Chapter 8
The Steady Magnetic Field
Curl
 To complete our original examination of the application of
Ampere’s circuital law to a differential-sized path, we may write
 H z H y 
 H x H z 
curl H    H  


ay
 ax  
z 
x 
 z
 y
 H y H x 


 az  J
y 
 x
H  J
• Point Form of Ampere’s circuital law
• Second Maxwell’s Equation,
non-time varying condition.
E  0
• Point Form of Potential Difference law
• Third Maxwell’s Equation,
non-time varying condition.
President University
Erwin Sitompul
EEM 11/13
Chapter 8
The Steady Magnetic Field
Homework 10
 D8.4.
 D8.5.
 All homework problems from Hayt and Buck, 7th Edition.
 Due: Monday, 01 July 2013.
President University
Erwin Sitompul
EEM 11/14