PHY 133
Formulas and Tables
You may refer to this handout on quizzes and exams. (You may also refer to the 131 and 132 handouts
if you find it necessary). Do not add additional information
k'
Things you should know from prerequisite coursess. (If you don’t, learn them now.)
- Definitions of trig functions
- Definition of pressure
- Pythagorean theorem
- Relationship between period and frequency
- Newton’s 3 laws
- Relationship between a wave’s frequency,
- Relationship between weight & mass
wavelength and speed
- PHY 131 formula for kinetic energy
- Definition of electric potential
- Formula for gravitational potential energy
- Formula for momentum
- Formula for angular momentum of a particle
- Conservation Laws (E, ⃑ & ⃑ )
Sec. 1:
Index of Refraction: n = c / v c = speed of light in a vacuum, v = speed of light in the material
Snell's Law:
Law of Reflection:
Magnification: M 
hi
s
 i
ho
so
________
From Phy 131 sec.11:
Harmonic wave traveling to the right: y = A sin (kx – ωt + )
where k = 2π/λ (wave number)
and ω = 2πf (angular frequency)
_
- 2speed of a string wave: v 
speed of sound in air =
F

F = string tension
μ = mass per unit length
402(T O K ) m/s ≈ [331.6 + .6 (TC)] m/s near 0C ≈ 343 m/s at 20C
Intensity: I = power/area_______________________________________________________________
Sec. 2:
Standing wave: y = A sin kx cos ωt
N to N = ½λ N to A = ¼λ
A to A = ½λ
Boundary condition for resonating string: node at a fixed end.
Air columns: Node at closed end, antinode at open end.
Fourier's Theorem: y(t) = Σ(An sin ωnt + Bn cos ωnt)
n
where fn = nf1
Beat frequency: fb = f1 - f2
f1 & f2 = frequency of the interfering waves____________________________________
Sec. 3:
Constructive interference if path difference = mλ
m = 0,1,2…
Destructive interfence if path difference = (m + ½)λ
Double-slit: Bright at d sinθ = mλ
Dark at d sinθ = (m + ½)λ
slit separation
Wavelength in material of index n: λn = λv/n
(λv = λ in vacuum)
Thin film interference:
n higher or lower than
n between
what's on either side:
what's on either side:
destructive: 2nt = mλv
destructive: 2nt = (m + ½)λv
constructive: 2nt = (m + ½)λv
constructive: 2nt = mλv
(t = thickness)
____________________________________________
Sec.4:
Diffraction grating: Bright at mλ = d sinθ
d = slit separation
Complete darkness at all other θ's
m = 0,1,2,
Bragg Equation: mλ = 2 d sinθ
θ = angle between beam & crystal planes
m = 1,2...
d = distance between crystal planes
Single slit:
Dark at: a sinθ = mλ
a = slit width
Bright at: θ = 0 and where a sinθ = (m+½)λ
m = 1,2,3,
- 3Angular separation of barely resolved sources:
Rectangular opening θ = λ
Circular opening: θ = 1.22 λ
a width
diameter 
D
Intensity transmitted by polarizing film: I = Io cos2θ
(θ = angle between ⃑ of incident light and transmission axis.)
Brewster's Law: n = tan θp
θp = incident angle for complete polarization
Sec. 5: Know the speed of light in a vacuum (to one significant figure).
_
Velocity Transformation:
Lorentz-Fitzgerald Contraction: Lp = L
Time Dilation: t = tp
Generalized Newton’s 2nd Law: ⃑
⃑
⃑
⃑
Total energy: E = γmc2
Rest energy: ER = mc2
Kinetic energy: KE = E - ER
________________________________________________________________________________
Sec. 6: Know formula for energy of a photon.
3
Wein’s Displacement Law: MAX T  2.898  10 m  K
Planck Radiation Law:
(
)
Max. Energy of Photoelectrons:
(
⁄(
)
KEMAX = eV0 = hf - 


Stopping Potential
DeBroglie Wavelength:
)
Work Function
p = momentum
Heisenberg Uncertainty Principle:
Also,
(
)
- 4SEC. 7:
Bohr’s Quantization Condition: mvr = n
Radius of H or H-like Atom: r = ( ) n2
ao =
n = 1,2,3, …
= Bohr Radius = .5292 Å
Spectral Series: Lyman: nf = 1,
Balmer: nf = 2,
Z = atomic number
Paschen: nf = 3,
Shells: n = 1, 2, 3, 4, …
K L M N…
Effective Atomic Number  Z-1 for K & L Electrons (n = 1 & 2),
SEC. 8:

2
Brackett: nf = 4,
Pfund:nf = 5
Z-9 for M Electrons (n = 3)____
(  dV = probability particle is in volume dV).
2
= probability density
One Dimensional, Time Independent Schrödinger Equation:
+
U = potential energy
E = energy
__________________________________________________________________________________
SEC. 9:
3 Dimensional Square Well:
 n x   n yy   n zz 
 sin
  A sin x  sin

 L   L   L 
nx = 1, 2, 3,…
ny = 1, 2, 3,…
nz = 1, 2, 3,…
E=
(
)
H Atom:
Principal Quantum No.: n = 1, 2, 3,…
Orbital Quantum No.:  = 0, 1, 2, …, (n-1)
Magnetic Quantum No.: m  0,  1,  2, ...,  
Spin Quantum No.:
ms = ± ½
subshells:
 = 0, 1, 2, 3, 4, …
s p d f g
- 5-
Spin: Z angular momentum: Sz = ms
e
-24
Z magnetic moment: μz = 
S z = 9.27 x 10 J/T
me
___________________________________________________________
Sec. 10: Maxwell’s Equations: Have a conceptual understanding.
EM wave propagating in x direction: ⃑ = ⃑ max cos (kx - ωt) ĵ
⃑ = ⃑ max cos (kx - ωt) k̂
(k = 2π/λ ω = 2πf)
c = speed of light in vacuum:
c
Emax E

Bmax
B
c
EM wave energy density: u = εo E2, uave = ½ εo Emax2
(energy per unit volume)
Poynting Vector: ⃑
⃑
⃑
1
0 0
(Erms = Emax/ 2 )
(in direction of propagation)
Intensity: I = |Sav| =
energy = power
(area) (time)
area
Waves's momentum: p = U/c (U = wave's energy)
Perfect absorber:
gains momentum = U/c
feels pressure = S/c
SEC. 11&12:
Pauli Exclusion Principle:
Perfect reflector:
gains momentum = 2U/c
feels pressure = 2S/c_____________________________________
No two electrons can be in the same state.
Hund’s Rule: In orbitals of equal energy, e-s are usually arranged for most unpaired spins.
From PHY 131: Rotational kinetic energy = ½Iω2,
Elastic potential energy = ½kx2
Diatomic Molecules:

Reduced Mass:
m1 m2
m1  m2
Moment of Inertia: I = r 2
- 6Allowed Rotational Energies =
(J)(J + 1)
(So, since ΔJ =  1 , ΔE = EJ – EJ – 1 =
J = 0, 1, 2, …
J)
Allowed Vibrational Energies = (v + ½) hf
v = 0, 1, 2, …
(So, since v  1 , ΔE = Ev – Ev – 1 = √ )
k = stiffness
f 
1
2
k

Fermi-Dirac Distribution Function (probability that a state of energy E has an e- in it):
f(E) =
)⁄
(
≈ 1 if E significantly < EF, ≈ 0 if E significantly > EF.
EF = Fermi Energy
k = Boltzmann Constant
Density of States (No. of States Per Unit Volume with Energy Between E and E + dE):
g(E)dE = CE½dE
3
8 2m 2
C
h3
= 1.062 x 1056 J-3/2·m-3 = 6.812 x 1027 ev-3/2·m-3 for electrons
Number of e-‘s per unit volume with energy between E & E + dE:
N(E)dE = f(E)g(E)dE
Number of e-s per unit volume:
√
)⁄
(
( )
∫
2
h 2  3n  3
Fermi Energy (at T = 0): EF 
 
8m   
Nuclear Radius:
r = r0A1/3
Fermi Velocity:
r0 = 1.2 fm, A = atomic mass
______________________________________________________________________________________________
Sec.13:
Radioactive Decay:
Number of Nuclei Present:
N = N0e  t
 = decay constant,
Decay Rate (or “activity”):
| |,
Units: SI unit: 1 Becquerel = 1 Bq = 1 decay/sec. 1 Curie = 1 Ci = 3.7 x 1010 Bq
Half-Life:
- 7-
Disintegration Energy/ Reaction Energy:
Q = (total m before – total m after)c2 ,
Radiation Dose Unit:
c2 = 931.5
(1 rad = .01 gray)
( J of energy absorbed, kg of mass absorbing it.)
Effective Dose in rem = (Dose in rad)(RBE)
RBE = Relative Biological Effectiveness
Sec. 14:
(1 rem = .01 sievert)
_
Baryon Number: For Baryons, B = 1; Antibaryons, B = -1; anything else, B = 0
Lepton Numbers:
For e- & e, Le = 1;
e+ &  e, Le = -1;
anything else, Le = 0
For  - &   , L = 1;
+ &  , L = -1;
anything else, L = 0
For  - &   , L  = 1;
 + &   , L  = -1; anything else, L  = 0
QUARK COLORS:
Red
Green
Blue
Particle
+
K+
KK0
QUARKS
u
d
c
s
t
b
LEPTONS
ee




COMPOSITION OF SEVERAL HADRONS
Mesons
Baryons
p
Quark Composition
n
̅
0
̅
+
̅
0
̅
̅
0
(A ∑0 is an excited state of the same quarks as in a Λ0.)
Hubble’s Law:
v = HR
where H 
⁄
uud
udd
uds (see below)
uus
uds (see below)
dds
uss
dss
sss
- 8Geometric Formulas:
(r = radius, h = height)
Circumference of a circle or sphere . . . . . . . . . . 2πr
Area of circle. . . . . . . . . . . . . . . . . . . . . . . . .. . . . πr2
Area of a circular cylinder (excluding ends) . . . .2πrh
Area of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . .4πr2
Volume of a circular cylinder . . . . . . . . .. . . . . . . .πr2h
Volume of a sphere . . . . . . . . . . . . . . . . . . . . . . . .(4/3)πr3
Quadratic Formula:
x
If ax2 + bx + c = 0 then
 b  b 2  4ac
2a
Short Table of Integrals: (a = some constant.)
 xn dx = xn+1 + c if n=/ -1
n+1
 sin(ax)dx = _ 1 cos(ax) + c
a
 dx/x = ln x + c
 cos(ax)dx = 1 sin(ax) + c
a
 eaxdx = 1 eax + c
a
 sin(ax)cos(ax)dx = sin2(ax) + c
2a
 xeaxdx = (ax-1) eax + c
a2
 sec2(ax)dx = 1 tan(ax) + c
a
 x2eaxdx = (a2x2-2ax+2) eax + c
a3
 csc2(ax)dx = _ 1 cot(ax) + c
a
 sin2(ax)dx = x - sin(2ax) + c
2
4a
 cos2(ax)dx = x + sin(2ax) + c
2
4a
Partial Derivatives:
Differentiate with respect to one variable, treating the others as constants.
example:
z = x + x2 y3
z = 1 + (2x)y3
x
z = 0 + x2(3y2)
y
- 9Some Fundamental Constants:
Speed of light in a vacuum. c = 2.998 x 108 m/s
Electron mass . . . . . . . . . . me = 9.110 x 10-31 kg = (.5110 MeV)/c2
Proton mass . . . . . . . . . . . mp = 1.673 x 10-27 kg = (938.3 MeV)/c2
Neutron mass . . . . . . . . . . mn = 1.675 x 10-27 kg = (939.6 MeV)/c2
Elementary charge . . . . . . . e = 1.602 x 10-19 C
Permittivity of free space . . εo= 8.854 x 10-12 c2/Nm2
1/4πεo = 8.988 x 109 Nm2/c2
Permeability of free space .
μo = 4π x 10-7 N/A2
Planck's constant . . . . . . . . h = 6.626 x 10-34 Js = 4.136 x 10-15 eVs
= h/2π = 1.055 x 10-34 Js = 6.582 x 10-16 eVs
Boltzmann's constant . . . . . k = 1.381 x 10-23 J/K = 8.617 x 10-5 eV/K
Avogadro’s number . . . . . . NA = 6.022 x 1023 particles/mole
_____________________________________________________________________________
Substance: Index of refraction: Substance: Index of refraction:
Solids:
Liquids:
Diamond . . . . . . 2.419
Benzene. . . . . . . . . . . 1.501
Fluorite . . . . . . . 1.434
Carbon disulfide . . . . 1.628
Fused quartz . . . 1.458
Carbon tetrachloride . 1.461
Glass, crown . . . 1.52
Ethyl alcohol . . . . . . . 1.361
Glass, flint . . . . . 1.66
Glycerine . . . . . . . . . . 1.473
Ice . . . . . . . . . . . 1.309
Water. . . . . . . . . . . . . 1.333
Polystyrene. . . . . 1.49
Gases (0C, 1 atm):
Sodium chloride 1.544
Air. . . . . . . . . . . . . . . 1.000293
Zicron . . . . . . . . 1.923
Carbon dioxide . . . . . 1.00045
_____________________________________________________________________________
SI Prefixes:
Prefix symbol factor
Yotta
Y
1024
kilo
k
103
nano
n
10-9
Zetta
Z
1021
hecto h
102
pico
p
10-12
Exa
E
1018
deka
da
101
femto
f
10-15
Peta
P
1015
deci
d
10-1
atto
a
10-18
Tera
T
1012
centi
c
10-2
zepto
z
10-21
Giga
G
109
milli
m
10-3
yocto
y
10-24
Mega
M
106
micro μ
10-6
- 10Units
Fundamental Units:
Standard SI Unit
and Abbreviation:
LENGTH
meter = m
TIME
second = s
MASS
kilogram = kg
CURRENT
ampere = C/s = A
TEMPERATURE kelvin = k
Conversion Factors
1 m = 3.28 ft, 1 mile = 1609 m = 5280 ft
1 Angstrom = 1 Å = 10-10 m, 1 inch = 2.54 cm
1 hour = 3600 s, 1 day = 86,400 s
1 year = 3.16 x 107 s
1kg = .0685 slug, weighs 2.21 lb in standard gravity
1 atomic mass unit = 1 u = 1.66 x 10-27 kg
T (in kelvins) = T (in Celsius) + 273.15
Derived Units
VOLUME
m3
1 Liter = 10-3 m3 = 10+3 cm3
SPEED
m/s
1 mi/hr = 0.447 m/s = 1.47 ft/sec
FORCE
newton = N
1 N = 0.225 pound
ENERGY
& WORK
joule = J
1 calorie = 4.186 J, 1 J = 0.738 ft·lb
1 electronvolt = 1.602 x 10-19 J
POWER
watt = W
1 horsepower = 746 W = 550 ftlb/sec
ANGLE
radian = rad
1 revolution = 360 = 2π rad
FREQUENCY
hertz = Hz
1 Hz = 60 rev/min = 1 cycle/sec
IMPULSE &
MOMENTUM
kgm/s = NS
CHARGE
coulomb = C
POTENTIAL
volt = V
DECAY RATE
becquerel = Bq
1 curie = 3.7 x 1010 Bq