u §6.5 electricity and magnetism - uprmece.uprm.edu/~pol/pdf/maxwell.pdfdr. s. cruz-pol, inel...
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Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Electrical Engineering, UPRM (please print on BOTH sides of paper) 1
Electricity and Magnetism INEL 4151
Sandra Cruz-Pol, Ph. D. ECE UPRM Mayagüez, PR
Topics u Electric Fields, [Coulomb’s Law], Gauss’ Law, E, D, V)
u Convection/conduction current, conductors, Polarization in dielectrics, Permittivity, conductors,[§5.3-5.5] resistance, capacitance [§6.5]
u Magnetic fields [Biot Savart Law], Ampere’s Law, u Flux Density, Magnetic Potentials , [§7.2-7.5, 7.7] u Magnetic Force, torque, moment, dipole, inductors, u Magnetic circuits [§8.2-8.3, 8.5-8.6, 8.8, 8.10]
Faradays Law, Transformer & Motional emf,
u Electromagnetic Waves: Maxwell Eqs., time varying potentials and Time Harmonic fields [§9.2-9.7]
u waves in different media, power and Poynting vector, u Incidence at normal angles. [§10.2-10.8]
u Transmission lines: Parameters, equations, Input impedance, SWR, power,
u Smith Chart [§11.2-11.5]
http://ece.uprm.edu/~pol/cursos
Some terms Ø E = electric field intensity [V/m] Ø D = electric field density or flux Ø H = magnetic field intensity, [A/m] Ø B = magnetic field density, [Teslas]
mH
mF
HBED
o
o
/1043610/1085.8
7
912
−
−−
×=
=×=
=
=
πµπ
ε
µ
ε
Vector Analysis Review: Ø What is a vector? Ø How to add them, multiply, etc,? Ø Coordinate systems
l Cartesian, cylindrical, spherical Ø Vector Calculus review
Vector Ø A vector has magnitude and direction.
Ø In Cartesian coordinates (x,y,z):
zzyyxx AAA aaaA ++=!
where aA is unit vector.
aA =!A!A=
!AA
222
aaa
zyx
zzyyxx
AAA
AAA
++
++=
!A= aAA
Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Electrical Engineering, UPRM (please print on BOTH sides of paper) 2
Vector operations Commutative
Associative
Distributive
kk AA!!
=ABBA
ABBA!!!!
!!!!
⋅=⋅
+=+
( )( ) BCACBAC
BABA!!!!!!!
!!!!
⋅+⋅=+⋅
+=+ kkk
A)()A(!!
kllk =C)BA()CB(A!!!!!!
++=++
Example Given vectors A=ax+3az and B=5ax+2ay-6az
Ø (a) |A+B| Ø (b) 5A-B
Answers: (a) 7 (b) (0,-2,21)
Vector Multiplications Ø Dot product
Ø Cross product
zyx
zyx
zyx
BBBAAAaaa ˆˆˆ
BA =×!!
nABaAB ˆsinBA θ=×!!
ABAB θcosBA =⋅!!
zzyyxx BABABA ++=⋅BA!!
22AAA A==⋅!!!
Note that:
Also… Ø Multiplying 3 vectors:
Ø Projection of vector A along B:
)BA(C)CA(B)CB(A
)BA(C)AC(B)CB(A!!!!!!!!!
!!!!!!!!!
⋅−⋅=××
×⋅=×⋅=×⋅
BaAAB ⋅=!!
Scalar:
Vector:
Example Given vectors A=ax+3az and B=5ax+2ay-6az
Ø (c) the component of A along y Ø (d) a unit vector parallel to 3A+B
Answers: (c) 0 (d) ± (0.9117,.2279,0.3419)
Coordinates Systems Ø Cartesian (x,y,z) Ø Cylindrical (ρ,φ,z) Ø Spherical (r,θ,φ)
Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I
Electrical Engineering, UPRM (please print on BOTH sides of paper) 3
Cylindrical coordinates (ρ,φ,z)
φρφρ
φρ
sincos
tan 122
==
=+= −
yxxyyx
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
z
y
x
z
zz
y
x
AAA
AAA
AAA
AAA
1000cossin0sincos
1000cossin0sincos
φφ
φφ
φφ
φφ
φ
ρ
φ
ρ
!A= Aρaρ + Aφ aφ + Azaz
ρ = Aρaρ + Aϕ aϕ + Azaz
Spherical coordinates, (r,θ,φ)
θφθφθ
φθ
cossinsincossin
tantan 122
1222
rzryrxxy
zyx
zyxr
===
=+
=++= −−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−−=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
z
y
xr
r
z
y
x
AAA
AAA
AAA
AAA
0cossinsinsincoscoscoscossinsincossin
0sincoscossincossinsinsincoscoscossin
φφ
θφθφθ
θφθφθ
θθ
φφθφθ
φφθφθ
φ
θ
φ
θ
φφθθ aaaA AAA rr ++=!
Vector calculus review
Del (gradient)
Divergence
Curl
Laplacian (del2 )
zA
yA
xA zyx
∂
∂+
∂
∂+
∂
∂=⋅∇ A!
zyx
zyx
AAAzyx
aaa
∂
∂
∂
∂
∂
∂=×∇
ˆˆˆ
A!
2
2
2
2
2
22
zV
yV
xVV
∂
∂+
∂
∂+
∂
∂=∇
zyx az
ay
ax
ˆˆˆ∂
∂+
∂
∂+
∂
∂=∇
Cartesian Coordinates
Theorems Ø Divergence
Ø Stokes’
Ø Laplacian
∫∫ ⋅∇=⋅vS
dvSd AA!!!
( )∫∫ ⋅×∇=⋅SL
Sdld!!!!
AA
zzyyxx aAaAaA
zV
yV
xVV
ˆˆˆA 2222
2
2
2
2
2
22
∇+∇+∇=∇
∂
∂+
∂
∂+
∂
∂=∇
!
Scalar:
Vector:
Vector calculus review Del (gradient)
zyx az
ay
ax
ˆˆˆ∂
∂+
∂
∂+
∂
∂=∇
∇ =∂∂ρ
aρ +1ρ∂∂φ
aφ +∂∂zaz
∇ =∂∂rar +
1r∂∂θ
aθ +1
rsinθ∂∂φ
aφ
In Othe
r
Coordin
ate
systems