Download PHYSICS 208 Exam 3/Final Exam: Summer 2009 Formula/Information Sheet

PHYSICS 208 Exam 3/Final Exam: Summer 2009
Formula/Information Sheet
• Basic constants:
Gravitational acceleration
Permittivity of free space
Coulomb constant
Permeability of free space
Elementary charge
Unit of energy: electron volt
Unit of energy: kilowatt-hour
Planck’s Constant
• Properties of some particles:
Proton
Electron
Neutron
g
0
k = 1/4π0
µ0
e
1 eV
1 kWh
h
=
=
=
=
=
=
=
=
9.8 m/sec2
8.8542 × 10−12 C2 /N·m2
8.9875 × 109 N·m2 /C2
4π × 10−7 T·m/A [ km = µ0 /4π = 10−7 Wb/A·m]
1.60 × 10−19 C
1.60 × 10−19 J
3.6 × 106 J
6.626 × 10−34 J sec
mass =1.67 × 10−27 kg
mass =9.11 × 10−31 kg
mass =1.67 × 10−27 kg
charge = + 1.60 × 10−19 C
charge = − 1.60 × 10−19 C
charge =0
• Some indefinite integrals:
R
R
R
dx
x
dx
(x2 +a2 )3/2
√ dx
x2 ±a2
=
=
ln x
√x
=
ln (x +
a2
R
R
x2 +a
√2
x2 ± a2 )
R
dx
a+bx
x dx
(x2 +a2 )3/2
√x dx
1
b
=
=
ln (a + bx)
− √ 21 2
√ x +a
x2 ± a2
=
x2 ±a2
• Basic equations for Electromagnetism:
Maxwell equations:
H
~ ~
H E · dl
~ ~
B
H · dl
~ · dA
~
E
H
=
=
=
=
~ · dA
~
B
B
− dΦ
dt
E
µ0 (ic + 0 dΦ
)
dt
1
Q
0 enclosed
0
• Basic Equations for Waves, Interference and Diffraction:
Wave Equation
Plane EM wave traveling in the +x direction
Speed of an EM wave [m/s]
Wave length of an EM wave [m]
Wave number of an EM wave
Poynting vector [J/s·m2 ]
Time-averaged S [J/s·m2 ]
Intensity of an EM wave [J/s·m2 ]
Total energy of an EM wave [J]
Total momentum of an EM wave
Law of Reflection
Snell’s Law
Lens Equation
Lens Maker’s Equation
Magnification
Double Slit Constructive Int.
Double Slit Destructive Int.
Intensity Maxima
Energy of an EM Wave (photon)
Single Slit Dest. Int.
∂ 2 f (x,t)
∂x2
E(x, t)
B(x, t)
c
λ
k
~
S
Save
I
U
|~
p|
θincident
n1 sin(θ1 )
1
f
1
f
m
d sin(θ)
d sin(θ)
I
φ
E
sin(θ)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
2
1 ∂ f (x,t)
v2
∂t2
Em cos(kx
− ωt)
Bm cos(kx − ωt)
Em
√ 1
= B
=
µ0 0
m
c
f
2π
λ
1 ~
E×
µ0
Em Bm
2µ0
~
B
Save
IAt
U
c
θreflected
n2 sin(θ2 )
1
1
+ s0
s
(n − 1)( R11 −
y0
= −s0
y
s
mλ
(m + 12 )λ
Io cos2 (φ/2)
2π
(r2 − r1 )
λ
hf
mλ
a
1
)
R2
E(x,t)
B(x,t)
• Basic equations for Electric Fields:
~|
|F
=
(point charge q)
~
E(r)
=
(group of charges)
(r̂ = unit
~
E
=
(continuous charge distribution)
~
E
~
(on q in E)
(r̂ = unit
~
F
=
Coulomb’s law
Electric field [N/C = V/m]
Electric force [N]
Electric flux
(through a small area ∆Ai )
∆Φi
=
(through an entire surface area)
Φsurf ace
=
=
~ i · ∆A
~ i = Ei ∆Ai cos θi
E
Z
lim
(through a closed surface area)
≡
Φclosed
X
∆A→0
I
Gauss’ law
|q1 ||q2 |
2
qr
k 2 r̂
r
vector
from q)
X radiallyX
qi
~
r̂
Ei = k
2 i
r
Z
i
dq
k
r̂
r2
vector radially from dq)
~
qE
k
~ · dA
~
E
∆Φi =
~ · dA
~ = Qin
E
0
Z
Electric potential [V = J/C]
(definition)
∆V
=
B
VB − VA = −
~ · d~l
E
A
~ = constant)
(E
(point charge q)
∆V
V (r)
=
=
(group of charges)
V (~r)
=
(continuous charge distribution)
V (~r)
=
~ · (~rB − ~rA )
−E
q
(with V (∞) = 0)
k
Xr
X
Vi (|~r − ~ri |) = k
(ViZ(∞) = 0)
dq 0
k
|~r − ~r 0 |
(V (∞) = 0)
Z
Electric potential energy [J]
(definition)
∆U
=
qi
|~r − ~ri |
B
UB − UA = −q0
A
~ · d~l
E
= q0 (VB − VA )
~
~
E
= −∇V,
~ = gradient operator can be expressed î(∂/∂x) + ĵ(∂/∂y) + k̂(∂/∂z)
where ∇
q1 q2
Electric potential energy of two-charge system
U12
= k
r12
~ from V
E
(definition)
C
≡
(parallel-plate capacitance)
C
=
Dielectrics in capacitors
C
=
Electrostatic potential energy [J] stored in capacitance
UE (t)
=
Capacitance [F]
Energy density [J/m3 ] in an Electric field
u
=
Electric dipole moment (2a = separation between two charges)
Torque on electric dipole moment
Potential energy of an electric dipole moment
|~
p|
~τ
U
=
=
=
Q
|∆V |
0 A
K
d
KCo
1 Q(t)2
2 C
1
o E 2
2
2aq
~
p
~×E
~
−~
p·E
• Basic equations for current and resistance:
Current [A]
Current density [A/m2 ]
Resistivity [Ω·m]
Resistance [Ω]
(definition)
with motion of charges
(definition)
for uniform cross-sectional area A
Energy loss rate on R [J/s]
Time constant in RC circuit [s]
Charging an RC circuit [q(t)]
Discharging an RC circuit [q(t)]
I
I
J
ρ
R
R
P
τRC
q(t)
q(t)
d Q(t)
dt
≡
=
=
=
≡
=
=
=
=
=
nqvd A
R
I
(where I = J~ · ~n dA)
A
~
|E|
~
|J|
V
I
ρ A`
2
I R = V 2 /R = IV
RC
qf (1 − e−t/RC )
qo e−t/RC
• Basic equations for Magnetism and Induction:
Magnetic force [N]
cyclotron motion
Magnetic moment [A·m2 or J/T]
Torque [N·m]
Energy [J]
on charge q
~
F
=
on current-carrying conductor
~
F
=
or
cyclotron radius
cyclotron frequency
F
r
ω
µ
~
~τ
U
=
=
=
=
=
=
~
B
=
on a current loop or magnet
of a current loop or magnet
Field of moving charge
I
Ampere’s law
~ · d~l
B
Magnetic field [T]
a long straight wire
inside a toroid
inside a solenoid
a straight wire segment
a circular arc (radius R)
~
dB
~
|B|
~
|B|
~
|B|
~
|B|
~
|B|
Displacement current [A]
(definition)
id
Biot-Savart law
=
=
=
=
=
=
=
≡
Ampere-Maxwell law
H
~ · d~s
B
=
Faraday’s Law
Self Inductance [H]
Self Induced electromotive force [V]
Mutual Inductance [H]
Electromotive force induced by mutual induction [V]
Magnetic field energy density
Magnetic energy stored in L [J]
Time constant in LR circuits [s]
Energizing an LR circuit [I(t)]
De-energizing an LR circuit [I(t)]
Angular frequency of LC circuit [rad/sec]
E
L
E
M21
E2
umagnetic
UB (t)
τLR
I(t)
I(t)
ω
=
=
=
=
=
=
=
=
=
=
=
Magnetic force [N]
Magnetic field [T]
Magnetic moment [A·m2 or J/T]
field due to sheet charge σm = qm /A
(definition)
(definition)
on pole qm
due to pole Qm
for poles ± qm separated by l
~
F
~
|B|
|~
µ|
~
|B|
=
=
=
=
~
qZ~v × B
~
Id~l × B
IlB sin θ
mv
qB
qB
m
~
N IA
~
µ
~ ×B
~
−~
µ·B
µo q ~v × r̂
4π
r2
µ0 I
µo I d~l × r̂
4π
r2
µ0 I/(2πa)
µ0 N I/(2πr)
µ0 N I/`
km I(cos θ1 − cos θ2 )/a
km Iθ/R
dΦE
0
dt
µ0 (ic + id )
− d dΦtm
N Φm
I
−L dd It
N2 ΦI21
1
−M21 ddIt1
1
B2
2 µo
1
LI(t)2
2
L
R
If (1 − e−tR/L )
−tR/L
Ip
oe
1
LC
~
qm B
km |Qm |
R2
qm l
2πkm |σm |