bütün fizik formülleri
TRANSCRIPT
1. Classical Mechanics - 20%
v2 = v20 + 2a(x− x0) v in terms of x in uniform acceleration
L = L(q, q, t) = T − U Lagrangianddt
(∂L∂q
)= ∂L
∂qEL eqs of motion
S =∫ t1t0L(q(t), q(t), t)dt Action, minimize this integral
p = ∂L∂q
generalized/conjugate momentum
F = ∂L∂q
generalized force
H = pq − L = T + U Hamiltonianq = ∂H
∂pp = −∂H
∂qH eqs of motion
p+ ρgy + 12ρv2 = constant Bernoulli’s eq
Fcoriolis = 2mr×Ω Coriolis forceFcentrifugal = −mω × (ω × r) Centrifugal forceτ = r× F torqueL = r× p angular momentumI =
∑imir
2i =
∫r2dm =
∫r2ρ dV moment of inertia
ω =√
mgrI
ω =√
g`
frequency of pendulum
τ 2 = 4π2
GMsa3 Kepler’s third law
where a is dist from center to aphelionI = Icm +md2 parallel axis theorem(
cos θ − sin θsin θ cos θ
)2D rotation matrix
µ = m1m2
m1+m2reduced mass
Ueff(r) = U(r) + `2
2µr22 body effective potential (1D)
ω =√
km
frequency of harmonic oscillator
v = ω × r v in terms of omega and r (in vectors)Fb = ρfV g bouyant forcefbeat = |f1 − f2| beats per second for 2 superposed wavesNo energy conserved in inelastic collision?
cs =√γ Pρ
γmono = 5/3 γdia = 7/5 speed of sound
γP = B = −∆p/(∆V/V ) definition of bulk modulusf = cs
λfrequency of sound
cs =√
γkTmper particle
= 343 m/s for air speed of sound of ideal gas (and for air)
cs =√γgh+ γ(atm)
ρspeed of sound in water
A1v1 = A2v2 pipe flow (non-viscous)
v =√
Tµ
speed of waves on a string
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R = Dvν
= ρDvη
Reynolds number
where D=char length, v=char speedη =dynamic viscosity, ν = kinematic viscosityR < 1 laminar, R > 1 turbulentdPdz
= −ρg hydrostatic equilibriump/p0 = ρ/ρ0 = e−z/H H = p0
gρ0= 8.55 km pressure vs height
M d~vdt
=∑Fext + ~vrel
dMdt
rocket equationv2 = 2GM
Rescape velocity
F = ηAdvdy
viscous force
λn = 2Ln
n = 1, 2, 3 . . . standing waves in open tube (fixed string)λn = 4L
nn = 1, 3, 5 . . . standing waves in closed tube (one end fixed
string)P = IV = Fv power definitionsLtotal = Lglobal + Lintrinsic total angular momentum of a systemstable non-circular orbits exist only in simple har-monic and inverse-square-force potentials
Bertrand’s theorem
ω =√w2
0 − t2
4m2 frequency of underdamped SHO
Moments of inertia:
Iz = mr2
2solid disc or cylinder
I = 2mr2
5solid sphere
I = 2mr2
3hollow sphere
Icenter = mL2
12rod about center
Icenter = mL2
3rod about end
kepler orbits?normal modes?
2. E&M - 18%
ε = “permittivity” µ = “permeability”ε = ε0(1 + χe) electric susceptibilityµ = µ0(1 + χm) magnetic susceptibilityε = ε0εr dielectric constant/relative permittivity
n =√
εµε0µ0
= cvp
=√εrµr index of refraction
c = 1√ε0µ0
speed of light
v = 1√εµ
v in terms of EM stuff
Typical values:
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ε > ε0 almost always, with χe > 0 permittivityµ < µ0, χm < 0 diamagneticsfield diesµ > µ0, χm > 0 paramagneticsµ >> µ0, χm >> 0 ferromagnetics∫
(∇ ·A)dτ =∮
A · da divergence theorem∫(∇×A) · da =
∮A · dl curl theorem
Maxwell Eqs:∇ · E = 1
ε0ρ Gauss’s Law
∇× E = −∂B∂t
Faraday’s Law∇ ·B = 0 No monopoles∇×B = µ0J + µ0ε0
∂E∂t
Ampere’s Law
∇ ·D = ρfree Gauss’s Law in matter∇×H = Jfree + ∂D
∂tAmpere’s Law in matter
D = ε0E + P definition of electric displacement DH = 1
µ0B−M definition of auxiliary field H
In linear media:P = ε0χeE polarization fieldD = εE displacement fieldM = χmH magnetization fieldH = 1
µB auxiliary field
V = N dΦdt
Induced voltage from Faraday’s LawF = 1
4πε0
q1q2r2
Coulomb’s Law
dB = µ0
4πIdL×rr2
Biot-Savart Law
B = µ0I2
r2
(r2+z2)3/2Magnetic field on axis of a circle of current
B = µ0I2πR
Magnetic field from infinite straight wiredF = I d`×B Force on a wire from a magnetic field
Boundary conditions:E⊥above − E⊥below = 1
ε0σ BCs for E
E‖above = E
‖below
D⊥above −D⊥below = σf BCs for D
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D‖above −D
‖below = P
‖above −P
‖below
B⊥above = B⊥below BCs for B
B‖above −B
‖below = µ0(K× n)
H⊥above −H⊥below = −(M⊥above −M⊥
below) BCs for H
H‖above −H
‖below = Kf × n
Q = CV U = 12CV 2 Capacitance and energy
C = εAd
Parallel plate capacitorV = Vb − Va = −LdI
dtU = 1
2LI2 Inductance and energy
Φ = LI inductance
L = µN2Al
Inductance of a solenoidZ = R Z = 1
iωCZ = iωL impedance for resistor, capacitor, inductor
ω = qBm
Cyclotron frequencycharged particle passes through a medium fasterthan v = c/n and emits blue light in nuclear re-actors
Cherenkov radiation
P = q2a2
6πε0c3Larmor formula for radiated power
ρ = EJ
resistivity
R = ρLA
resistance, L = length, A = areaJ = σE conductivity10−4 m/s electron drift velocityJ = I
A= σE = nevdrift equation for drift velocity
E‖ = 0 outsiede, ~E reversed upon reflection reflection off a conductor
E ∝ −µ0p0ω2
4π( sin θ
r)θ E field of oscillating electric dipole
B ∝ −µ0p0ω2
4π( sin θ
r)φ B field of oscillating electric dipole
E ∝ µ0m0ω2
4πc( sin θ
r)φ E field of oscillating magnetic dipole
B ∝ −µ0m0ω2
4πc( sin θ
r)θ B field of oscillating magnetic dipole
p = qd electric dipole momentm = IA magnetic dipole momentσ = −ε0
∂V∂n
surface charge on conductor vs VZload = Z∗source impedance matchingσb = P · n bound surface chargeρb = −∇ ·P bound volume chargeKb = M× n bound surface currentJb = ∇×M bound volume currentused to measure exponent in Coulomb law placeda charge inside a hollow charged sphere measuredhow far it moved (not much)
Cavendish-Maxwell experiment
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τ = µ×B torque on a dipoleU = −µ ·B dipole potential energyB = µ0µ
2πz3magnetic dipole field
L = µ0N2h2π
ln ba
inductance of a toroidE = 1
4πε0
px3 electric dipole field (approx)
E = 14πε0
p(x2+(d/2)2)3/2
electric dipole field (exact)
current flows in metal in perp B field, charge accu-mulates on one side, magnetic force balances newE field, determines sign of charge carriers
Hall effect
S = 1µ0
(E×B) Poynting vector
gauges?EM field transformations ?cross product identities?
3. Optics and Wave Phenomena - 9%
vphase = ωk
vgroup = dωdk
Phase velocity versus group velocitygroup velocity information travels at...
ω = ω01±vobs/cn
1∓vsource/cnDoppler shift (for sound waves in a medium)
Top signs moving towards each other1o
+ 1i
= 1f
Thin lens formula
sin θ = ∆lf≈ 1.22λ
dRayleigh Criterion (aperture)
f = R2
1f
= (n− 1)(
1R1
+ 1R2
)Focal length of mirror and lens
I0 cos2(θ − φ) Intensity of polarized light through polarizerym = mλD
ddouble slit interference, position of maxima
d sin θ = mλ m = 0,±1,±2, . . . same equation in terms of θw sin θ = mλ m = ±1,±2, . . . single slit diffraction, position of minimaIθ = 4I0 cos2(πd sin θ
λ) Iθ for 2 slit interference
Iθ = Imax( sinαα
)2 Iθ for 1 slit diffractionwhere α = πa
λsin θ
product of the above 2 slit interference and diffractionreflected wave same phase wave at boundary with n1 > n2
reflected wave 180 phase shift wave at boundary with n1 < n2
λ = vν
wavelengthobject side image sidetan θB = n2
n1Brewster’s angle
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θB measured from vertical to incident raycomponent perp to surface is eliminated2d sin θ = nλ condition for constructive interference for Bragg
diffractiond is distance between latticesθ is angle from horizontal
mθ = θeyθob
= −fobfey
magnification for telescope
λn = λn
wavelength in materialreference wave interferes with wave reflected offobject, stores phase and amplitude info
hologram
path diff = 2d+ 12λn for each phase shift thin films interference
sin θ = λd
diffraction, angle to edge of central maxminimize blur = d+ dscreen using above eqn pinhole camera sharpness
4. Thermodynamics and Statistical Mechanics - 10%
e = WQinput
definition of efficiency
1− Tc/Th efficiency of Carnot cycle∆U = ∆S = 0 Carnot cycleadiabat steeper than isotherm on PV plotrectangle on US plotZ =
∑s e−Es/kT P (s) = e−Es/kT/Z Boltzmann distribution
U , N , and V entropy is a function of...S = kB ln Ω Approximate definition of entropyS = −kB
∑s ps ln(ps) Exact definition of entropy
∆S = Nk lnVfVi
change in entropy when U and N are fixed
Q =∫TdS relation between entropy and heat
12kT per quad dof Equipartition theoremλmaxT ≈ 3× 10−3m ·K Wien’s Law (for a blackbody)j∗ = σT 4 power per unit area radiated by blackbodyσ = 5.67× 10−8 Stefan-Boltzmann constantni = gi
e(εi−µ)/kT+1Fermi-Dirac stat, number fermions in state i
ni = gie(εi−µ)/kT−1
Bose-Einstein stat, number bosons in state i
g degeneracy, ε energy,µ chem potential ∂U
∂N
PV = NkT ideal gas lawk = 1.381× 10−23J/K Boltzmann’s constant3 dofs for monatomic ideal gas
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3 at > 10K dofs for cold diatomic ideal gas5 at > 100K dofs for diatomic ideal gas7 at > 1000K dofs for hot diatomic ideal gas
vmax =√
2kTm
vmax for an ideal gas
v =√
8kTπm
v for an ideal gas
vrms =√
3kTm
vrms for an ideal gas
Uthermal = N · f · 12kT total thermal energy of ideal gas
∆U = Q+Won sys 1st law of thermodWon sys = −PdV work in quasistatic compressionQ = 0 adiabaticT = const isothermalW = NkT ln Vi
Vfwork in isothermal compression of ideal gas
V Tf2 = const V
f+2f P = const relations for adiabatic compression
t1/2 = ln 2k
half lifewhere k = decay ratenumber of indep events in some time Poisson distribution√N uncertainty of N-count P distribution
σ =√< (x− x)2 > standard deviation
aka RMS deviation from meanσ2 varianceC = heat needed
degree temp increaseheat capacity
c = C/m specific heat
CV = (∂U∂T
)V = Nfk2
heat capacity at constant volumeCP = (∂U
∂T)P + P (∂V
∂T)P = CV +Nk heat capacity at constant pressure
A = Pρw
scattering cross section
where P is probability and w is width∆L = αL∆T linear expansionenergy of highest QM state at T = absolute zero Fermi energy∆Q∆t
= −kA∆T∆x
Fourier’s law (heat conduction)
5. Quantum Mechanics - 12%
Time-Independent Perturbation TheoryE1n = 〈Ψ0
n|H ′|Ψ0n〉 first order correction to energy
|Ψ1n〉 =
∑m 6=n
〈Ψ0m|H′|Ψ0
n〉E0n−E0
m|Ψ0
m〉 first order correction to wavefn
E2n =
∑m6=n
|〈Ψ0m|H′|Ψ0
n〉|2E0n−E0
msecond order correction to energy
degeneracies?
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i~∂Ψ∂t
= HΨ Ψ(t) = e−iHt/~Ψ(0) time dependent SEHΨ = EΨ Ψ(t) = e−iEt/~Ψ(0) time independent SE
H = − ~2
2m∇2 + V Hamiltonian
p = −i~∇ momentum operator[x, p] = i~ commutator of x and pσAσB ≥ | 1
2i〈[A,B]〉| generalized uncertainty principle
σxσp ≥ ~/2 Heisenberg uncertainty principleλ = h
p= 2π~
pde Broglie Wavelength
E = hc/λ energy in terms of wavelength|1, 1〉 =↑↑ triplet states (j = 1)|1, 0〉 = 1√
2(↑↓ + ↓↑)
|1,−1〉 =↓↓|0, 0〉 = 1√
2(↑↓ − ↓↑) singlet state (j = 0)
Pauli spin matrices:(0 11 0
)σx(
0 −ii 0
)σy(
1 00 −1
)σz
[AB,C] = A[B,C] + [A,C]B identity for [AB,C]P =
∫|Ψ|2dV =
∫Prdr how P relates to Pr
l = 0 wavefunctions with spherical symmetry?in scattering: incoming plane wave not substan-tially altered by potential
Born approximation
ψ(r1, r2) = A(ψa(r1)ψb(r2) + ψb(r1)ψa(r2)) bosonssym, do not obey Pauli exclusionPboson ∝ T 5/2
ψ(r1, r2) = A(ψa(r1)ψb(r2)− ψb(r1)ψa(r2)) fermionsantisym, obey Pauli exclusion
E = n2π2~2
2ma2 energy levels in square well
ψn(x) =√
2a
sin(nπax) wavefunctions in square well
p = ~k p in terms of kψ0 = 1√
πa3e−r/a0 Hydrogen ground state wavefunction
H ′ = eEextz perturbed H for Stark effect
Might not need:d〈Q〉dt
= i~〈[H,Q]〉+ 〈∂Q
∂t〉
[Li, Lj] = i~Lkεijk angular momentum commutators
6. Atomic Physics - 10%
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Electron quantum numbersn = 1, 2, 3, . . . principle quantum number, radial wavefn` = 0, 1, . . . , n− 1 orbital quantum number, θ wavefnL2ψ = ~2`(`+ 1)ψm` = −`,−`+ 1, . . . , `− 1, ` magnetic quantum number, φ wavefnLzψ = ~m`ψs = (1/2) electron spinS2|ψ〉 = ~2s(s+ 1)|ψ〉ms = ±(1/2) z-component of electron spinSz|ψ〉 = ~ms|ψ〉j = |`− s|, . . . , (`+ s) = |`± (1/2)| total ang momentum quantum numberJ2|ψ〉 = ~2j(j + 1)|ψ〉mj = −j,−j + 1, . . . , j − 1, j z-component of total angular momentummj = m` +ms Jz|ψ〉 = ~mj|ψ〉∆` = ±1 ( 6= 0) ∆m` = 0,±1 electric dipole transition selection rules∆j = 0,±1 ∆ms = 0Wigner=Eckart theorem governs theseenergy level splitting when ext B field applied Zeeman effectelectrical analog of Zeeman effect Stark effect
an = 4πε0~2
µe2Zn2 ≈ (0.529A)
(meµ· n2
Z
)Bohr model radius
En = Z2µe4
8ε20h2
(−1n2
)≈ (−13.6 eV)
(Z2
n2 · µme
)Bohr model energy
Eγ = −13.6eV( 1n2i− 1
n2f) energy of radiated photon from transition
1λ
= R( 1n2f− 1
n2i) wavelength of radiated photon
R = 1.097× 107m−1 Rydberg constant
x ray spectrum from electrons fired at atoms:photon knocks our inner e−, replaced by outer e− Auger transitioncontinuous spectrum, radiation produced by de-celeration of a charged particle
Bremsstrahlung
K, L, M, N → nf = 1, 2, 3, 4 x ray series namespick L (S,P,D,F,G,H,... for L = 0, 1, 2, . . .) term symbolwhere J is between L+ S and |L− S|write spin multiplicity = 2S+1LJµ = γS = (−eg/2me)S intrinsic magnetic moment
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g = 2 γe = 1.76× 1011 gyromagnetic ration for an electronphotons hitting metal knocks off electrons, lightshines on emitter, sending electrons to a collector,making a current, which stops at some negativevoltage (at V=0 some electrons still collected)
Photoelectric effect
dominant process up to 500 KeVminimum energy needed to free electron from ma-terial
work function
photon scatters off electron, changes wavelength,electron flies off
Compton scattering
∆λ = hmc
(1− cosφ) Compton effect equationφ is angle between scattered photon and horizon-talconfirmed Bohr’s quantized model, atoms couldonly absorb specific amounts of energy
Franck-Hertz experiment
dσdΩ
= ( kqQ2mv20
)2 1sin4(θ/2)
Rutherford scattering
σ incident ring, Ω scattered solid angleshot α particles at gold foil, found concentratednucleus
σ = ( q2
4πε0mc2)2 Thomson scattering
σ is the cross section (area/solid angle)photons scattering off electronsphoton destroyed, e+ e− created, pair productionbecomes significant at 10 MeVNsc = Nincntarσ classical scattering relationNsc(into dΩ) = Nincntar
dσdΩ
(θ, φ)dΩ diff scat cross (experiment)dσdΩ
= bsin θ| dbdθ| diff scat cross (theory)
σ is cross section area of one targetΩ = A/r2 definition of solid angledΩ = sin θdθdφgamma ray photons from nuclear transitionx ray photons from electron transitionnucleus interacts with lower shell electron, eject-ing it, higher shell electron fills hole, x ray whenheavy nucleus, visible when H atom
internal conversion
λ = 10−10 ν = 1018 x rayλ = 10−12 ν = 1020 gamma ray
orbitals: go diagonal top right to bottom left
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quantum numbers: row denotes n, col denotes lnumber of electrons a shell can hold determinedby ms and ml
s p d f g1 1st2 2nd 3rd3 4th 5th 7th4 6th 8th 10th 13th5 9th 11th 14th 17th 21st
Might not need:anomalous Zeeman effect?hyperfine splitting?
Enj = −13.6eVn2 [1 + α2
n2 ( nj+1/2
− 34)] energy levels of H w/ fine structure
7. Special Relativity - 6%
γ = 1√1−v2/c2
≥ 1 Lorentz factor
γ ≈ 1 + 12v2
c21γ≈ 1− 1
2v2
c2
ct′ = γ (ct− βx) x′ = γ (x− β(ct)) Lorentz transformations
L′ = Lproper
γlength contraction
T ′ = (Tproper)γ time dilation
ω = ω0
√1∓v/c1±v/c doppler shift for light
p = γmv relativistic momentum
E =√
(mc2)2 + (pc)2 = γmc2 relativistic energyblack hole if R = 2GM
c2Schwarzchild radius
ω = ω0
γ(1−β cos θ)general doppler shift
vAC = vAB+vBC1+(vABvBC/c2)
velocity addition
C is lab frame
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8. Laboratory Methods - 6%
∆f = (∑n
i=1[(∆xi)(∂if(x1, . . . , xn))]2)1/2
error propagation∆ff
=√∑
(∆xixi
)2 error propagation if no cross-corellated terms
1 J ≈ 6× 1018 eV Joules to eV conversion(kB)(300 K) ≈ .02 eV room temp in eVλ = 750− 380 nm ν = 400− 790× 1012 Hz visible light in m, Hz, eVE = 2− 3 eV10−6m/s drift velocity of electrons in metalfunctions like y = ax straight lines on semilog plotfunctions like y = xa straight lines on log log plotτc = RC ≈ 63% RC time constant
9. Specialized Topics - 9%
QM version of normal modes, vibration in lattice phononcollection of Fermions, QM ideal gas Fermi gas3 quarks (such as p, n), hadron baryoneach quark = 1/3 baryon numberνe, e, νµ, µ, ντ , τ leptonse, µ, τ each have own number lepton numberlepton and corresp ν have +1, anti have -12 quarks (such as π+ pion, K− kaon), hadron mesonEbind/c
2 = (total mass of nucleons) −(mass of nucleus)
binding energy
defined as positive, corresponds to negative po-tential energyiron element with highest binding energy per nucleon2 protons, 2 neutrons α particlen→ p+ e− + νe β− decayp→ n+ e+ + νe β+ decayN-doped, P-doped semicond with more electrons, holesρ = ρ0(1 + α∆T ) resistivity as a fn of ∆Tα > 0 α for conductorsα < 0 α for semiconductorsquarks, leptons weak force affectsquarks, gluons, hadrons strong force affectscrystal lattice looks the same from every point Bravais lattice
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can cover entire lattice, contains 1 point primitive unit cellsum density of each point (1/number sharedblocks)
calculate lattice point density
2 body centered point density4 face centered point densityproton and neutron, boson deuteronNo weak interactions preserve strangeness?
10. RandomdAdt
= −kA⇔ A = A0e−kt
sin θ ≈ tan θ when θ is small
1.414√
2
1.732√
3sin 2θ = 2 sin θ cos θ trig identitiescos 2θ = 2 cos2 θ − 1
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