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Final Notesheet-1

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ECE 390 Final Notesheet
Electrostatic
−
→
−
→
Force on an Electric Charge: F Q1 = Q1 E Q2
Electrostatic in Materials (Maxwell’s
Equations):
‹
−
−
→
ρ
− →
→
Q
∇· E =
⇐⇒
E · ds = enc
Eo
Eo
S
‹
−
→
− →
→
−
∇ · D = ρf ree
⇐⇒
D · ds = Qf ree
→
−
∇× E =0
˛S
⇐⇒
→
−
− →
→
−
D = Eo E + P
→
→ −
−
E · dl = 0
(Gauss’ Law)
(Gauss’ Law)
(KVL)
C
Electrostatic in Linear Isotropic Materials (Dielectric):
−
→
−
→
→
−
−
→
P = Eo x e E
D = Eo Er E
Er = 1 + x e
−
→
→
Dipole Moment: −
p =Qd
Q
Capacitance: C =
∆Φ
Electric Field
− V
→
E
m
Point Charge
Line Charge
Plane Charge
Q
R̂
4πEo Er R2
ρl
R̂
2πEo Er R
ρs
ẑ
2Eo Er
Arbitrary Condition
ˆ
ρl
R̂ dl
4πEo Er R2
C
¨
ρs
R̂ ds
4πEo Er R2
S
˚
ρ
R̂ dv
4πEo Er R2
V
Displacement Field
− C oul
→
D
m2
ρs
ẑ
2
ρl
R̂
2πR
Q
R̂
4πR2
ˆ
C
¨
S
˚
ρl
R̂ dl
4πR2
ρs
R̂ ds
4πR2
ρ
R̂ dv
4πR2
V
ˆ
Electric Potential
J
Φ∞ [V ] or
C
Q
4πEo Er R
ˆ
C
ρl
dl
4πEo Er R
¨
S
ρs
ds
4πEo Er R
C
¨
S
˚
V
ρl
dl
4πEo Er R
ρs
ds
4πEo Er R
ρ
dv
4πEo Er R
Electric Potential:
ˆP2
− →
→
−
∆Φ = − E · dl = Φ2 − Φ1
P1
ˆr
Φ∞ = −
→ −
−
→
E · dl = −
∞
ˆr
Er dr
∞
→
−
E = −∇Φ
Parallel Plates:
−
→
ρ
ρs
E = s
∆Φ = d
Eo
Eo
Eo A
d
C=
Coaxial Cable:
For a ≤ r ≤ b,
Assume l >> r
→
−
E =
Q
r̂
2πEo rl
b
Q
∆Φ =
ln
a
2πEo l
2πEo l
Q
C=
=
∆Φ
ln ab
Parallel Wires:
For 0 ≤ r ≤ d,
Assume d >> a
→
−
ρl
ρl
E = −
+
x̂
2πEr x
2πEo (x − d)
∆Φ =
ρl
πEo
ln
d
a πEo
C
= ρl =
l
∆Φ ln
d
a
l
l
∆Φ
= ρsh
=
A
W
I
1
Sheet resistivity: ρsh =
σt
Resistance: R = ρ
Conductivity:
−
→
−
→
−
→
dρ
J = σE
∇· J =−
dt
‹
I =
→ →
−
−
J · ds
τ = RC =
S
Steady State Conduction:
−
→
∇· J =0
⇐⇒
∇×
→
−
J
σ
‹
S
=0
⇐⇒
˛
C
→ −
−
→
J · ds = 0
→
−
J
σ
−
→
· dl = 0
(KCL)
(KVL)
Eo Er
σ
Magnetostatic
−
→
ˆ
−
→
I dl × R
Biot-Savart Law: d H =
− 2
→
4π| R |
Maxwell’s Equation:
−
→
∇· B =0
⇐⇒
→ →
−
−
∇×H = J
‹
˛S
⇐⇒
→ −
−
→
B · ds = 0
→ −
−
→
H · dl = I
(Ampere’s Law)
C
(→
→
−
− −
→\
B = µo H + M
→
−
− →
→
−
With: D = Eo E + P
In Free Space:
−
→
−
→
p =0
M =0
−
→
−
→
−
→
−
→
D = Eo E
B = µo H
−
→
−
→
→
Magnetic Dipole Moment: −
m = I A = qm d
Infinitely Long Line
Current Sheet
I
φ̂
2πR
JS
φ̂
2
Magnetic Field
− A
→
H
m
Solenoid
JS φ̂
NI
L
Arbitrary Current
ˆ →
−
I dl × R̂
− dl
→
4π| R |2
¨C
JS × R̂
A
− 2 ds m
→
4π| R |
S
˚ →
−
J × R̂
A
− 2 dv m2
→
4π| R |
V
Susceptibility and Permeability:
−
→
−
→
M = xH
−
→
−
→
B = µo µr H
µr = 1 + x
Inductance:
L =
Impedance:
Zo =
Φ
N
I
(where I = JS W for current through parallel plates)
L
C
Magnetic
Circuits: (Assume flux is uniform and contained inside permeable material)
˛
− →
→
−
H · dl = N I
φ
B = µo µr H =
A
N I = φ�
l
�=
µ oµ r A
φ: flux
�: reluctance
A: cross sectional area
Symbols and Units for Basic Quantities
Symbol
Quantity
Unit/Value
−
→
B
Magnetic flux density
[T esla] or
−
→
D
Electric flux intensity, Displacement field
C
m2
−
→
E
Electric field intensity
−
→
F
Force
−
→
H
Magnetic field intensity
I
Current
−
→
J
Current density
−
→
JS
Surface current density
L
Inductance
−
→
M
Magnetization vector
Amps
m
−
→
m
Magnetic dipole moment
Amps
m2
−
→
P
Electric polarization vector
Wb
m2
V olts
m
[N ]
Amps
m
[Amps]
Amps
m2
Amps
m
[H ]
C
m2
−
→
p
Electric dipole moment
Q
Electric charge
[C ]
qe
Electron charge
1.602 × 10
R
Resistance
[Ω]
RS
Surface resistance
[Ω]
Zo
Characteristic impedance
[Ω]
a
Bohr radius
5.2918 × 10
Eo
Permittivity of free space
8.854 × 10
Er
Relative permittivity
[unitless]
µo
Permeability of free space
4π × 10
µr
Relative permeability
[unitless]
ρ
Volume charge density
ρs
Surface charge density
C
m2
ρl
Line charge density
C
m
σ
Conductivity
[S]
Φ
Electrostatic potential
[V olts] or
C
m
−19
C
−11
−11
m
F
m
−7 H
m
C
m3
J
C
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