Download ρV volume charge density (C/m3)

Symbols and Constants
Q
F~
~
E
~
D
Φ
V
~
H
~
B
Φ
~
S
Z
P
N
charge (C)
force (N)
electric field (V/m)
displacement field (C/m2 )
electric flux (C)
electric potential (V)
magnetic field (A/m)
magnetic flux density (T)
magnetic flux (Wb)
surface vector (m2 )
impedance (Ω)
power (W)
number of turns (coil)
ρV
ρS
ρL
0
R
W
I
µ0
µR
volume charge density (C/m3 )
surface charge density (C/m2 )
linear charge density (C/m)
8.854 × 10−12 F/m
dielectric constant
energy (J)
current (A)
4π × 10−7 Wb/Am
relative permeability
Y
T
admittance (S)
period (s)
Mathematics
spheres
volume = (4/3)πa3
area = 4πa2
cylinders
volume = πa2 L
area: (cross section) πa2 , (side wall) 2πaL
complex numbers
ejθ = cos θ + j sin θ
for z = α + jβ
|z| = (α2 + β 2 )1/2
and
6
θ = tan−1 (β/α)
Electric fields
Coulomb’s law
Q1 Q2
F~2 =
R̂12
4π0 R2
point charge
~ =
E
electric flux
~ ·S
~
Φ=D
linear dielectric
~ = 0 R E
~
D
charge density
Q = ρV × volume
Guass’s law
for a closed surface Φ = Qencl
~ to V
relating E
Q
r̂
4π0 r2
VAB = −
~
F~ = QE
V =
Q
4π0 r
~
(for flat surface and constant D)
Z A
B
Q = ρS × area
~
~ in the same direction as x)
|E|dx
(for E
C=
0 R A
(parallel plate cap.)
d
capacitance
C = Q/V
energy
1~ ~
~
WE = E
· D × volume (for constant E)
2
W = QV
Magnetic fields
long straight wire
~ = I φ̂
H
2πr
solenoid
~ = nI ẑ
H
Faraday’s law
emf = −
self-inductance
L = N Φ/I (from linked flux)
mutual-inductance
M = N2 Φ12 /I1 (from linked flux)
linear materials
~ = µ0 µR H
~
B
energy
1~ ~
~
WH = H
· B × volume (for constant H)
2
Lorentz force
~
F = q~v × B
(n is the number of turns per meter)
dΦ
dt
~
force on a straight wire F~ = I~l × B
sliding bar
~
V = l~v × B
where Φ is magnetic flux
L=
2WH
(from energy)
I2
~ constant along wire)
(l is the length; B
~ constant over loop)
(l is the length and ~v the velocity; B
Electric circuits
dV
(capacitor)
dt
I=C
stored energy
1
1
W = CV 2 (capacitor) W = LI 2 (inductor)
2
2
dissipated power
P = I 2 R = V 2 /R = IV
RL and RC transients
V (t) = V1 + V2 e−t/τ
(constant source)
V =L
dI
(inductor) V = IR (resistor)
dt
fundamental equations
I(t) = I1 + I2 e−t/τ
τ=
1
L
or
RC
R
Kirchoff’s laws
sum of the voltage drops around a loop equals zero
sum of currents into a node equals zero
Electric AC circuits
frequency
ω = 2πf
f=
Ohm’s law
V = IZ
impedance
Z=
1
(capacitor)
jωC
admittance
Y =
1
Z
combining impedance
series Z = Z1 + Z2
1
T
Z = R (resistor)
Z = jωL (inductor)
parallel Z = Z1−1 + Z2−1
−1
!1/2
1ZT 2
V (t) dt
(T is the period)
T 0
RMS
VRMS =
instantaneous power
P = Re(V )Re(I)
actual average power
P̄ =
phasor notation
zejωt ⇒ |z|6 θ
2
VRMS
cos θ where Z = |Z|ejθ
|Z|
power factor = cos θ
Magnetic circuits
coil
mmf = N I
material section
mmf = Hl
flux
Φ = BA
field to flux density
obtain from the magnetization curve for a material
B = µ0 µR H (µR = 1 for air)
reluctance
R=
Ohm’s law
mmf = ΦR
(l is the length)
(A is the cross-sectional area)
l
µ0 µR A
(l is the length and A is the cross-sectional area)