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Exams Summer 2022 > General Physics > Flashcards

General Physics Flashcards

1
Q

What is the maxwell - boltzmann distribution?

A

p(v) =~ v^2 e^( - m v^2 / 2kT )

Probability of a particle of mass m in a gas of temperature T having a velocity v.

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2
Q

What are the constituents of a lambda baryon?

A

u d s

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3
Q

What equation relates the thermal and electrical conductivity in a solid?

A

thermal conductivity / electrical conductivity ~ T

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4
Q

What is the equipartition theorem?

A

Quadratic terms in the classical expansion for energy contribute 1/2 kT per particle. (for sufficiently high T)

  • translational : 3/2 kT (for 3D)
  • rotational : 1/2 kT per axis (kT for diatomic)
  • vibrational : kT (KE and “elastic” potential)
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5
Q

What is the fermi-dirac distribution?

What about bose-einstein?

A

For fermions:
{state E} = 1 / ( e^[ (E - mu) / kT] - 1)

For bosons:
{state E} = 1 / ( e^[ (E - mu) / kT] + 1)

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6
Q

What are maxwell’s equations in differential and integral form?

A

Refer to notes

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7
Q

What are the dimensions of energy?

A

M L^2 T^-2

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8
Q

What are the dimensions of viscosity?

A

M L^-1 T^-1

energy density * time

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9
Q

What is the energy of a massless particle?

A

E = pc = hc / lambda = hf = hbar omega

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10
Q

What is the binomial distribution?

A

(look it up)

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11
Q

What is the key relation for an ideal heat engine?

A

Q1 / T1 = Q2 / T2

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12
Q

What is the definition of the poynting vector?

What is the relation between |B| and |E| for EM waves?

A

S = E x H
The poyting vector is the direction and magnitude of energy flux (W m^-2)

|B| = |E| / c

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13
Q

What is the relativistic doppler shift equation?

A

lambda’ / lambda = [ (c + v) / (c - v) ]^1/2

Combined effects of time dilation and stretching of the wavefront due to the motion.

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14
Q

How is angular velocity defined?

A 
v = omega r (tangential)
omega = 2 pi f
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15
Q

What is the equation for nuclear decay of a source?

A

N(t) = N_0 * e^(-lambda t)

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16
Q

For forced, damped SHM in the absence of driving, what is the amplitude response?

How can we work out the solution?

A

A = A_0 * e^( - gamma * t / 2)

Use classic DE method, auxiliary function gives complex: sum of ( cosine + sine {of complex part}) * exponential(of real part)

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17
Q

What is the relation for heat capacities Cv and Cp for ideal gases?

A

Cp - Cv = nR = Nk

Cv is the normal one

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18
Q

What is the lorentz force equation?

A

F = q(E + v x B)

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19
Q

How is the coefficient of friction defined?

A

The frictional force is equal to the coefficient of friction * the normal force.

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20
Q

What is the integral required for first-order perturbation theory?

A

simply the expectation value of the perturbing potential, evaluated using the ground state wavefunctions.

(normalised wavefunctions)

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21
Q

How to propagate the error for a function f(x, y,…)?

A

the variance = partial derivitave w.r.t. x squared * x variance + ….

The variables must be uncorrelated

Always use radians!

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22
Q

What is the defining equation of thermal conductivity?

A

Heat flux = - thermal_conductivity * delta T / d

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23
Q

What are the boundary conditions for E B D and H?

A

B perp and E para are continuous.

D perp and H para are continuous (no free surface charges/currents).

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24
Q

What is the equation for phase velocity?
What about group velocity?
What is the difference between them?

A

Phase velocity = omega / k
Group velocity = d (omega) / d (k)

Group velocity is the velocity of the wavepacket, whereas the phase velocity is the velocity of waves within the wave packet.

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25
Q

What is the equation for the photoelectric effect?

A

E_max = hc / lamda - work_function

E_max is the maximum emitted photon energy

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26
Q

What is the fundamental thermodynamic relation?

A

dE = TdS - PdV

dQ = TdS
( dW = -PdV )

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27
Q

When does constructive interference happen?

What happens to phase on reflection at a boundary with a less dense medium?

A

Constructive interference when the optical path difference is an integer number of wavelengths.

pi (half wavelength) phase difference on reflection at a boundary with a less dense medium.

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28
Q

What is the SHM equation?

A

a = - omega^2 x

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29
Q

When should D and H be used for maxwell’s equations?

A

Use them for the sourced equations.
The current and charge are now only the “free” versions.

D is continuous (E is not). H is NOT continuous (B is).

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30
Q

What is the expression for the coulomb force between two point charges?

A

q1 q2 / (4 pi e_0 r^2)

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31
Q

For canonical systems, how is the partition function defined?

A

The sum over states of energy E:

e^( - E / kT )

(the normalisation of the boltzmann distribution)

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32
Q

What are the dimensions of surface tension?

A

M T^-2

energy per unit area

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33
Q

What is the potential characteristic of a quantum harmonic oscillator?
What are the corresponding energy levels?

A

V = 1/2 m omega^2 x^2

E = hbar omega (n_1 + n_2 + .. + N/2
for N dimensional)

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34
Q

What is the diffraction limit?

A

For light passing through a circular aperture:

delta (theta) ~= 1.22 * lambda / d

d is the diameter of the aperture.
delta (theta) is the distance to the first minima from the central peak.

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35
Q

What is the critical temp of a bose-einstein condensate proportional to?

At what point do quantum effects need to be considered?

A

Tc =~ n^(-1/3)

Quantum effects when the atomic spacing is roughly the de-broglie wavelength.

–> n ~ 1 / (lamda)^3

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36
Q

How to approach problems involving orbits of two bodies?

A

Use COM frame (define r from COM):

m1 R1 = m2 R2

T1 = T2

equate circular motion acceleration and gravitational force, make sure to use the correct r’s.

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37
Q

What is the entropy change for a reversible change?

A

The entropy of the universe does not change.

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38
Q

What is the de-broglie relation for energy and momentum?

A 
lambda = h/p
E = hbar omega (KE only)
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39
Q

What are the equations for time dilation and event contraction?

A 
t = gamma * t_0
L = L_0 / gamma
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40
Q

If there is no information about the error on a set of n measurements, how should you estimate the standard deviation?

A

sqrt(n)

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41
Q

What integral can we use to find the work done by a component?

A

W = integral w.r.t. q {V}

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42
Q

What is the equation for the voltage across a capacitor?

What about across an inductor?

A

V = Q/C

V = L derivative w.r.t. t{ I }

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43
Q

What is the heisenberg uncertainty principle and the corresponding principle for energy width?

A

delta x * delta p >= hbar / 2

Gamma = delta E = hbar / tau
tau is the lifetime
(be careful when calculating this for wavelength line width)

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44
Q

How is torque defined?

A

r cross Force

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45
Q

If a set of n measurements Xi have equal error sigma:

  • what is the mean
  • what is the error on the mean
  • what is the variance
  • what is the estimate of the standard deviation
A

Estimate the mean as usual.
Error on mean = sigma / sqrt(n)
Variance = 1/n * sum of (Xi - mean)^2
Estimate of error on measurements: sqrt(variance) * sqrt(n / (n-1))

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46
Q

What is gamma? (relativity)

A

1 / sqrt(1 - beta^2)

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47
Q

What is the force due to a spring?

What about the tortional force?

A

spring:
F = -kx

tortion:
torque = - K theta

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48
Q

Describe a HR diagram.

A

x-axis: temp (high to low) or colour (blue to red) (log)
y-axis: luminosity (relative) (log)

main sequence in the middle \
white dwarfs bottom left
red giants top right and supergiants above them

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49
Q

What is meant by a distance of one parsec?

A

The distance at which a star will undergo parallax motion of 1 arcsecond as seen from the earth’s orbit of the sun (1AU)

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50
Q

What is the frequency of visible light?

A

red light has ~ 4 x 10^14 Hz

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51
Q

What are the energy levels of a hydrogen-like potential?

A

E = - Z^2 * E_R / n^2

Z is the atomic number
E_R is the rydberg energy, use REDUCED MASS
n = 1, 2, 3…

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52
Q

What is the lens formula?

A

1/u + 1/v = 1/f

u and v are define in opposite directions.

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53
Q

What are the constituents of a K+ meson?

A

u sbar

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54
Q

What is the rotational counterpart to newtons second law?

A

torque = the rate of change of angular momentum

= moment of inertia * rate of change of angular velocity

55
Q

What are the constituents of a K0 meson?

A

d sbar

56
Q

What is Wein’s displacement law?

A

omega_(peak) = 2.82 kT / hbar

57
Q

What is the expectation value of momentum for bound states?

A

0

58
Q

What is a partial pressure?

A

For a mix of ideal gases, the pressure is equal to the sum of partial pressures, and the ideal gas equations apply to each partial pressure separately.

59
Q

What is luminosity?

A

Total power emitted by the object

60
Q

How is rotational energy defined?

A

1/2 * I * omega^2 = 1/2 * (L^2)/I

61
Q

What are the names of the angular momentum states of a hydrogen-like potential?

A

s, p, d, f…

0, 1, 2, 3…

62
Q

What are the constituents of a sigma baryon?

A

uus

63
Q

What are the components of four-momentum?

A

(gamma m c, gamma m v) = (E/c, p)

64
Q

What is the equation for a geometric sum ( a * r^n ) from n=0 to infinity?

A

a / ( 1 - r )

65
Q

What is the DE for forced, damped SHM?

A

x’’ + gamma x’ + omega_0 ^2 x = F/m

66
Q

What are the dimensions of force?

also tension

A

M L T^-2

67
Q

Would a bell with little damping (it rings out for a long time) have a low or high Q - factor?

A

High Q - factor, gamma is much less than the resonant frequency,

68
Q

What is the momentum operator (position representation)?

A

-i hbar d/dx

partial

69
Q

What is the no-slip condition for a rotating body?

A

v=omega * r at the surface

70
Q

How can we find the most probable particle velocity from the maxwell - boltzmann distribution?

A

Where the derivative of the distribution w.r.t. v is 0.

71
Q

How can we find the average energy from the partition function?

A

= - ( partial derivative w.r.t. beta { ln(Z)})

= - 1/Z * partial derivative w.r.t. beta {Z}

72
Q

What is the equation for energy of a relativistic particle?

A

E^2 = (pc)^2 + (mc^2)^2 = gamma * mc^2

73
Q

What equation should be used for the pressure in a deep fluid?

A

P + intergral w.r.t. depth (density * g) = constant

74
Q

How can we calculate the deviation of an observable?

A

(delta x)^2 = [x^2] - [x]^2

75
Q

What is the expression for EM energy density?

A

u = 1/2 * (D E + B H)

76
Q

What is an expression for heat capacity at const volume Cv, valid for all fluids?

A

partial derivative of E w.r.t. T, const V.

77
Q

What is the magnitude of the four-momentum?

A

PP -> M^2 c^2 = E^2 / c^2 - p^2

78
Q

What is the energy stored by a capacitor?

What about by an inductor?

A

1/2 * C V^2

1/2 * L I^2

79
Q

What is the definition of entropy in thermodynamics?

A

dS = dQ / T

Delta S = integral (dQ / T)

For phase changes there is no temperature change so Delta S = L / T

80
Q

What is the equation for beat frequency?

A

The beat frequency is equal to the difference between the two interfering frequencies.

81
Q

How is the helmholtz free energy defined for canonical systems?

A

F = -k T ln(Z)

82
Q

What are the units of electric field?

A

V m^-1

83
Q

What are the equations for diffraction and interference from slits far away from a screen?

A

a sin(theta) = n lambda

a = slit width for diffraction
= slit separation for interference

84
Q

What are the rotational counterparts to normal quantities?

A

mass –> moment of inertia
velocity –> angular velocity
momentum –> angular momentum
force –> torque

85
Q

What do isothermal, isobaric and isochoric mean?

A

Isothermal: no temperature change (no energy change for ideal gases)
Isobaric: no pressure change
Isochoric: no volume change (-> no work done)

86
Q

What are the eigenstates and energy levels for the 1D infinite potential well?

A

E_n = (hbar k_n)^2 / 2m

k_n = n pi / L

wavefunctions are sine functions, with root (2/L) normalisation.
Multiply the stationary states by e^-i (w_n t).

E_n = hbar w_n

87
Q

For forced, damped, SHM with sinusoidal driving force:

  • how to find the amplitude response?
  • when is critical damping?
  • what is meant by gamma?
  • What is the “Q” value?
A

-substitute complex number x_0 e^(i (wt) ), then solve. You will have to rationalise the denominator and take the real part.

  • Critical damping when gamma = 2 omega_0
  • Gamma is the FWHM of the power curve.

Q = omega_0 / gamma
Q is the “quality factor” of the oscillator

88
Q

How is angular momentum defined? (mechanics)

A

L = r cross p

Also L = I omega

89
Q

What are the constituents of a omega baryon?

A

sss

90
Q

What is the blackbody radiation equation?

A

Power (per unit area) = s-b constant * surface temp^4

= L / A

91
Q

What is the magnitude of the buoyant force?

A

Equal to the weight of fluid displaced.

92
Q

What is the mass number of nitrogen?

A

14

However, recall that nitrogen gas is diatomic.

93
Q

How is the moment of inertia defined?

A

The integral of r^2 dm, where r is defined from the pivot.

94
Q

How is GPE defined?

What is the equation for the gravitational attraction between two bodies?

A

the integral of the gravitational force between two points.

For a spherical body:
(GMm)/r

F = G (Mm)/r^2

95
Q

What is the range of values that m_j can take?

A

magnitude <= j

2j + 1 states

96
Q

What is the ideal gas equation?

What is the other key ideal gas property?

A

PV = NkT = nRT

E == E(T)

97
Q

What is the boltzmann distribution?

A

p(state E) =~ e^(- E/ k T )

Remember to normalise with the partition function

98
Q

What is the heat capacity of a solid (high T)?

A

3R

99
Q

What is the half-life?

What about the lifetime?

A

halflife = ln(2) / lambda = lifetime * ln(2)

lifetime = 1/ lambda

100
Q

What does adiabatic mean? What relation can we use for adiabatic expansion/compression?

A

Adiabatic: no heat transfer dQ = 0

P * V^gamma = constant

gamma = Cp / Cv

101
Q

What is the first law of thermodynamics?

A

The change in internal energy = the heat supplied to the system + the work done on the system

102
Q

How to work out the direction of cross products?

A

a x b

first finger - a
second finger - b
thumb points in direction of cross product

103
Q

What is the approximate molar mass of air?

A

29g

104
Q

What are the constituents of a K- meson?

A

ubar s

105
Q

What are the constituents of a K0bar meson?

A

dbar s

106
Q

If J = J_1 + J_2, what different j states can be formed?

A

j = j_1 + j_2 , … | j_1 - j_2|

107
Q

What is Bragg’s law?

A

For light (wavelength lambda) constructively reflecting off a crystal lattice (spacing d) at an angle theta to the surface:

n * lambda = 2d sin(theta)

[extra factor of 2 compared to diffraction or interference]

108
Q

What is the difference between an elastic and inelastic collision, and in which is there the maximal velocity transfer?

A

Elastic collision: momentum and KE conserved. (maximal velocity transfer)
Inelastic collision: momentum conserved

109
Q

What are two expressions for current density?

A

j = sigma E = n q v

n is number density, v is drift velocity
sigma is electrical conductivity

110
Q

What is fermat’s principle?

A

Light minimises the optical path (integral w.r.t. path of the refractive index)

111
Q

What is a typical separation of particles in a gas, liquid, solid?

A

Gas ~ 10 angstrom

Liquid/solid ~ 1 angstrom

112
Q

What are some suvat equations?

A 
v = u + at
s = ut + 1/2 a t^2
v^2 = u^2 + 2as
s = 1/2(u + v)t

and the same for the angular counterparts.

113
Q

What is the definition of entropy in statistical thermodynamics?

A

S = k ln(Omega)

where Omega is the #microstates corresponding to the macrostate

114
Q

What is the magnitude equation?
What is the difference between flux density and luminosity?
What kind of scale is the magnitude scale?

A

m2 - m1 = 2.5 * log{10} ( F1 / F2 )

flux density = luminosity / (4 pi r^2)

magnitude is an inverse logarithmic scale

115
Q

What is the equation for constructive thin-film interference?

A

(m+1/2) lambda = 2 d n cos (theta)

where theta is the angle of refraction

[cos instead of sine, factor of 2n and also factor of 1/2 due to phase change on reflection]

116
Q

What is the nonrelativistic equation for doppler shift?

A

(delta lambda) / lambda = v / c

(delta f) / f = v / c

117
Q

What is the defining equation for refractive index?

A

n = c/v = lambda(vacuo) / lambda(medium)

frequency is constant

118
Q

How are D and H defined?

A 
D = e_r e_0 E
B = mu_r mu_0 H
119
Q

What is the 3D density of states in k-space, and how can the equivalent for energy-space be found?

A

g(k) = ( V k^2 / 2 pi^2 ) * g(s)

use E = (hbar^2 k^2) / (2m)

120
Q

What is the reduced mass?

A

(Mm)/(M+m)

121
Q

What is the zeeman splitting potential (strong field)?

A

delta E = 1/2 * mu_B * B * (m_l + 2m_s)

122
Q

What is the eigenvalue of J^2 operator on an eigenstate?

What about the J_z operator?

A

hbar^2 j(j+1)

hbar m_j

123
Q

What is the equation for distance to an object using parallax?

A

distance (parsecs) = 1 / the parallax angle (arcseconds)

124
Q

What is the Taylor expansion equation?

A

–

125
Q

What are the lorentz transforms for a frame S’ moving in the positive x direction, with frames co-linear at t=0?

A 
ct' = gamma * (ct - beta * x)
x' = gamma * (x - beta * (ct) )
y' = y
z' = z

Other way round - switch signs.
Remember - dashed subject -> minus sign.
Be careful with the direction of the frame movement.

126
Q

What is the acceleration of a point undergoing circular motion?

A

a = - v^2 / r = - omega^2 r

127
Q

What is Malus’ law?

A

The intensity transmitted by a plane polariser:
I = I_0 * cos^2 (theta)

theta is the angle between the initial E-vector and the pass axis.

128
Q

What is the capacitance of a parallel-plate capacitor?

A

C = e_0 A/d

C = Q / V

129
Q

How to “motivate” the schwarzchild radius?

A

Equate classical KE and GPE (GMm/r), set v = c.

2 G M / c^2

130
Q

What is an expression for the mean free path (lambda) in a gas?
What is it important to note?

What is the probability of a collision happening in some time period?

A

mean free path = 1 / ( n sigma)

  • sigma is the collision cross section (pi (2r)^2)
  • n is the number density

Note that this does not play the same role as the decay constant in nuclear decay.

Probability = e^(- x / lambda)

131
Q

What is snell’s law?

A

n1 sin( theta1 ) = n2 sin( theta2 )

Angles are defined from the normal of the surface.

132
Q

What is the range of visible wavelengths of light?

A

400 to 800 nm

133
Q

What is brewster’s angle?

A

Theta = arctan (n_2 / n_1)

Upon incidence at this angle, light polarised in the plane perpendicular to the surface is not reflected at all.

Exams Summer 2022 (5 decks)

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