3 3 Curl and Divergence Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edw[.]
Trang 1Slide Presentations for ECE 329,
Introduction to Electromagnetic Fields,
to supplement “Elements of Engineering
Electromagnetics, Sixth Edition”
by
Nannapaneni Narayana Rao
Edward C Jordan Professor of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India
Trang 23.3 Curl and Divergence
Trang 3Maxwell’s Equations in Differential Form
Curl
Divergence
=
=
t t
B
×E
D
× H J
x y z
x y z
× Α
D
B
A
= x y z
A
Trang 4Basic definition of curl
× A
is the maximum value of circulation of A per
unit area in the limit that the area shrinks to the point
Direction of is the direction of the normal
vector to the area in the limit that the area shrinks
to the point, and in the right-hand sense
× A
max
Lim
d
A l
× A = a
Trang 5Curl Meter
is a device to probe the field for studying the curl of the field It responds to the circulation of the field
Trang 63.3-6
Trang 70
2 for 0
2 2
2
z
z
a
a
a v
a
negative for 0 2
positive for
2
y
a x
a x a
× v
0
0
2
2
y z
y
y z
v
v
a v
a
a
Trang 8Basic definition of divergence
Divergence meter
is the outward flux of A per unit volume in the limit that
the volume shrinks to the point
is a device to probe the field for studying the divergence
of the field It responds to the closed surface integral of the vector field
Lim
0
A d S A=
Trang 9At the point (1, 1, 0)
Divergence zero
Divergence positive
Divergence negative
(a)
(b)
(c)
2
y 1 ay
y
x
y a
y
1
y
1
x
y
1
z 1
x
Trang 10Two Useful Theorems:
Stokes’ theorem
Divergence theorem
A useful identity
C A d l = S × A d S
× A
S A d S = V A dv
Trang 11x y z
x y z
A A A
× Α