lecture 1 - introduction€¦ · 5 chapters 1, 2 and 3 in your text book! ab x x y y z z x x y y z...
TRANSCRIPT
Electromagnetics
EE 340
Dr. Mohammad S. SharawiEE, KFUPM
20121
Lecture 1 - Introduction
Why Study EM?
© 2012, M. S. Sharawi
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Because EM phenomena is in all electrical/electronic based equipment ….
Electrical/electronic components/equipment are almost everywhere …
Thus, EM is everywhere these days …
Computers, Cell Phones, Car Controllers, Power Lines, etc.
Wireless communications is based on EM wave propagation ...
High speed digital design is based on EM wave propagation …
Fields around power lines are EM fields …
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?
Vector Algebra Quick Review (MATH302!)
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Chapters 1, 2 and 3 in your text book!
zzyyxxAB
zzyyxx
zyx
zzyyxxA
zyx
zzyyxxA
BABABAABaAaAaA
AAA
aAaAaAA
a
AAAA
aAaAaAaA
++==•
•=•=•=
++
++===
++=
++==
θcos
ˆ , ˆ , ˆctor,another veor axisan on vector a of projection
ˆˆˆˆ
)(cartesian ˆˆˆˆ
222
222
BAAAA
AAA
A
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( ) ( ) ( ) zyx azzayyaxxRRR
ˆˆˆ 121212
1212
−+−+−=−=
Position Vector,
Example 1.1:Given a vector
find the magnitude of A, its unit vector and the angle that it makes with the z-axis? [On the Board.]
zyx aaa ˆ2ˆ2ˆ −+−=A
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Cross Product,
Scalar Triple product,ABBA
BA
×−=×
=×
=×
=×
==×
yxz
xzy
zyx
zyx
zyx
zyx
nAB
aaaaaaaaa
BBBAAAaaa
aAB
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆˆsinθ
( ) ( ) ( )
zyx
zyx
zyx
CCCBBBAAA
=
ו=ו=ו BACACBCBA
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Vector Triple Product,
( ) ( ) ( )BACACBCBA ••−••=××
A) Cartesian Coordinates (x,y,z)
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- already dealt with!
dxdydzdvadxdyadxdzadydzd
adzadyadxd
z
y
x
zyx
==
==
++=
ˆ
ˆ ˆ
ˆˆˆ
sl
B) Cylindrical Coordinates (ρ,φ,z)
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zz , siny , cos
, tan ,
ˆˆˆˆˆˆˆˆˆ
ˆˆˆ
122
222
===
=
=+=
=×
=×
=×
++=
++=
−
φρφρ
φρ
φρ
ρφ
φρ
φρ
φφρρ
x
zzxyyx
aaaaaaaaa
AAAA
aAaAaA
z
z
z
z
zzA
- useful for problems with cylindrical symmetry
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Transformation matrices
−=
−=
zz
y
x
z
y
x
z
AAA
AAA
AAA
AAA
φ
ρ
φ
ρ
φφϕφ
φφϕφ
1000cossin0sincos
1000cossin0sincos
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dzdddvadd
adzdadzdd
adzadadd
z
z
φρρφρρ
ρ
φρ
φρρ
φ
ρ
φρ
==
=
=
++=
ˆ
ˆ
ˆˆˆˆ
sl
C) Spherical Coordinate System (r,ϴ,φ)
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used in problems with spherical symmetry.
θφθφθ
φθ
θφ
φθ
φθ
φθρ
φφθθ
cosz , sinsiny , cossin
tan , tan ,
ˆˆˆˆˆˆˆˆˆ
ˆˆˆ
122
1-222
222
rrrx
xy
zyx
zyxr
aaaaaaaaa
AAAA
aAaAaA
r
r
r
rr
===
=
+=++=
=×
=×
=×
++=
++=
−
A
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Transformation Matrices,
−
−=
−−=
φ
θ
φ
θ
θθφφθφθφϕθφθ
φφθφθφθθϕθφθ
AAA
AAA
AAA
AAA
r
z
y
x
z
y
xr
0sincoscossincossinsinsincoscoscossin
0cossinsinsincoscoscos
cossinsincoscos
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φθθ
φφθφθθ
φθθθ
φ
θ
φθ
ddrdrdv
ardrdadrdr
addrd
addrardadrd
r
r
sin
ˆ ˆsin
ˆsin
ˆsinˆˆ
2
2
=
===
++=
s
l
Gradient of a Scalar Function
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A vector (magnitude and direction) that represents the maximum space rate of increase of a function A.
Gradient in other coordinate systems is given in your text book.
zyx azAa
yAa
xAA ˆˆˆ
∂∂
+∂∂
+∂∂
=∇
Divergence of a Vector and Divergence Theorem
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The divergence of A at a point P is the outward flux per unit volume as the volume shrinks about P.
The divergence theorem states that the total outward flux of a vector field A through the closed surface S is the same as the volume integral of the divergence of A.
zA
yA
xAA zyx
∂∂
+∂
∂+
∂∂
=•∇
∫∫ •∇=•VS
dvAsdA
Curl of a Vector and Stokes’s Theorem
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The curl of A is a rotational vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is normal to the direction of the area when oriented to make maximum circulation.
Stokes’s Theorem: The circulation of a vector field A around a closed path is equal to the surface integral of the curl of A over the open surface bounded by the path.
( ) AAcurl ×∇=
( )∫∫ •×∇=•SL
sdAldA
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More exercises in the book and during the first few lab sessions.
Next time …
© 2012, M. S. Sharawi
Do Not Forget to check the class page often …
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