General Physics Flashcards
What is the maxwell - boltzmann distribution?
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.
What are the constituents of a lambda baryon?
u d s
What equation relates the thermal and electrical conductivity in a solid?
thermal conductivity / electrical conductivity ~ T
What is the equipartition theorem?
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)
What is the fermi-dirac distribution?
What about bose-einstein?
For fermions:
{state E} = 1 / ( e^[ (E - mu) / kT] - 1)
For bosons:
{state E} = 1 / ( e^[ (E - mu) / kT] + 1)
What are maxwell’s equations in differential and integral form?
Refer to notes
What are the dimensions of energy?
M L^2 T^-2
What are the dimensions of viscosity?
M L^-1 T^-1
energy density * time
What is the energy of a massless particle?
E = pc = hc / lambda = hf = hbar omega
What is the binomial distribution?
(look it up)
What is the key relation for an ideal heat engine?
Q1 / T1 = Q2 / T2
What is the definition of the poynting vector?
What is the relation between |B| and |E| for EM waves?
S = E x H
The poyting vector is the direction and magnitude of energy flux (W m^-2)
|B| = |E| / c
What is the relativistic doppler shift equation?
lambda’ / lambda = [ (c + v) / (c - v) ]^1/2
Combined effects of time dilation and stretching of the wavefront due to the motion.
How is angular velocity defined?
v = omega r (tangential) omega = 2 pi f
What is the equation for nuclear decay of a source?
N(t) = N_0 * e^(-lambda t)
For forced, damped SHM in the absence of driving, what is the amplitude response?
How can we work out the solution?
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)
What is the relation for heat capacities Cv and Cp for ideal gases?
Cp - Cv = nR = Nk
Cv is the normal one
What is the lorentz force equation?
F = q(E + v x B)
How is the coefficient of friction defined?
The frictional force is equal to the coefficient of friction * the normal force.
What is the integral required for first-order perturbation theory?
simply the expectation value of the perturbing potential, evaluated using the ground state wavefunctions.
(normalised wavefunctions)
How to propagate the error for a function f(x, y,…)?
the variance = partial derivitave w.r.t. x squared * x variance + ….
The variables must be uncorrelated
Always use radians!
What is the defining equation of thermal conductivity?
Heat flux = - thermal_conductivity * delta T / d
What are the boundary conditions for E B D and H?
B perp and E para are continuous.
D perp and H para are continuous (no free surface charges/currents).
What is the equation for phase velocity?
What about group velocity?
What is the difference between them?
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.
What is the equation for the photoelectric effect?
E_max = hc / lamda - work_function
E_max is the maximum emitted photon energy
What is the fundamental thermodynamic relation?
dE = TdS - PdV
dQ = TdS
( dW = -PdV )
When does constructive interference happen?
What happens to phase on reflection at a boundary with a less dense medium?
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.
What is the SHM equation?
a = - omega^2 x
When should D and H be used for maxwell’s equations?
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).
What is the expression for the coulomb force between two point charges?
q1 q2 / (4 pi e_0 r^2)
For canonical systems, how is the partition function defined?
The sum over states of energy E:
e^( - E / kT )
(the normalisation of the boltzmann distribution)
What are the dimensions of surface tension?
M T^-2
energy per unit area
What is the potential characteristic of a quantum harmonic oscillator?
What are the corresponding energy levels?
V = 1/2 m omega^2 x^2
E = hbar omega (n_1 + n_2 + .. + N/2
for N dimensional)
What is the diffraction limit?
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.
What is the critical temp of a bose-einstein condensate proportional to?
At what point do quantum effects need to be considered?
Tc =~ n^(-1/3)
Quantum effects when the atomic spacing is roughly the de-broglie wavelength.
–> n ~ 1 / (lamda)^3
How to approach problems involving orbits of two bodies?
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.
What is the entropy change for a reversible change?
The entropy of the universe does not change.
What is the de-broglie relation for energy and momentum?
lambda = h/p E = hbar omega (KE only)
What are the equations for time dilation and event contraction?
t = gamma * t_0 L = L_0 / gamma
If there is no information about the error on a set of n measurements, how should you estimate the standard deviation?
sqrt(n)
What integral can we use to find the work done by a component?
W = integral w.r.t. q {V}
What is the equation for the voltage across a capacitor?
What about across an inductor?
V = Q/C
V = L derivative w.r.t. t{ I }
What is the heisenberg uncertainty principle and the corresponding principle for energy width?
delta x * delta p >= hbar / 2
Gamma = delta E = hbar / tau
tau is the lifetime
(be careful when calculating this for wavelength line width)
How is torque defined?
r cross Force
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
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))
What is gamma? (relativity)
1 / sqrt(1 - beta^2)
What is the force due to a spring?
What about the tortional force?
spring:
F = -kx
tortion:
torque = - K theta
Describe a HR diagram.
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
What is meant by a distance of one parsec?
The distance at which a star will undergo parallax motion of 1 arcsecond as seen from the earth’s orbit of the sun (1AU)
What is the frequency of visible light?
red light has ~ 4 x 10^14 Hz
What are the energy levels of a hydrogen-like potential?
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…
What is the lens formula?
1/u + 1/v = 1/f
u and v are define in opposite directions.
What are the constituents of a K+ meson?
u sbar
What is the rotational counterpart to newtons second law?
torque = the rate of change of angular momentum
= moment of inertia * rate of change of angular velocity
What are the constituents of a K0 meson?
d sbar
What is Wein’s displacement law?
omega_(peak) = 2.82 kT / hbar
What is the expectation value of momentum for bound states?
0
What is a partial pressure?
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.
What is luminosity?
Total power emitted by the object
How is rotational energy defined?
1/2 * I * omega^2 = 1/2 * (L^2)/I
What are the names of the angular momentum states of a hydrogen-like potential?
s, p, d, f…
0, 1, 2, 3…
What are the constituents of a sigma baryon?
uus
What are the components of four-momentum?
(gamma m c, gamma m v) = (E/c, p)
What is the equation for a geometric sum ( a * r^n ) from n=0 to infinity?
a / ( 1 - r )
What is the DE for forced, damped SHM?
x’’ + gamma x’ + omega_0 ^2 x = F/m
What are the dimensions of force?
also tension
M L T^-2
Would a bell with little damping (it rings out for a long time) have a low or high Q - factor?
High Q - factor, gamma is much less than the resonant frequency,
What is the momentum operator (position representation)?
-i hbar d/dx
partial
What is the no-slip condition for a rotating body?
v=omega * r at the surface
How can we find the most probable particle velocity from the maxwell - boltzmann distribution?
Where the derivative of the distribution w.r.t. v is 0.
How can we find the average energy from the partition function?
= - ( partial derivative w.r.t. beta { ln(Z)})
= - 1/Z * partial derivative w.r.t. beta {Z}
What is the equation for energy of a relativistic particle?
E^2 = (pc)^2 + (mc^2)^2 = gamma * mc^2
What equation should be used for the pressure in a deep fluid?
P + intergral w.r.t. depth (density * g) = constant
How can we calculate the deviation of an observable?
(delta x)^2 = [x^2] - [x]^2
What is the expression for EM energy density?
u = 1/2 * (D E + B H)
What is an expression for heat capacity at const volume Cv, valid for all fluids?
partial derivative of E w.r.t. T, const V.
What is the magnitude of the four-momentum?
PP -> M^2 c^2 = E^2 / c^2 - p^2
What is the energy stored by a capacitor?
What about by an inductor?
1/2 * C V^2
1/2 * L I^2
What is the definition of entropy in thermodynamics?
dS = dQ / T
Delta S = integral (dQ / T)
For phase changes there is no temperature change so Delta S = L / T
What is the equation for beat frequency?
The beat frequency is equal to the difference between the two interfering frequencies.
How is the helmholtz free energy defined for canonical systems?
F = -k T ln(Z)
What are the units of electric field?
V m^-1
What are the equations for diffraction and interference from slits far away from a screen?
a sin(theta) = n lambda
a = slit width for diffraction
= slit separation for interference
What are the rotational counterparts to normal quantities?
mass –> moment of inertia
velocity –> angular velocity
momentum –> angular momentum
force –> torque
What do isothermal, isobaric and isochoric mean?
Isothermal: no temperature change (no energy change for ideal gases)
Isobaric: no pressure change
Isochoric: no volume change (-> no work done)
What are the eigenstates and energy levels for the 1D infinite potential well?
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
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?
-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
How is angular momentum defined? (mechanics)
L = r cross p
Also L = I omega
What are the constituents of a omega baryon?
sss
What is the blackbody radiation equation?
Power (per unit area) = s-b constant * surface temp^4
= L / A
What is the magnitude of the buoyant force?
Equal to the weight of fluid displaced.
What is the mass number of nitrogen?
14
However, recall that nitrogen gas is diatomic.
How is the moment of inertia defined?
The integral of r^2 dm, where r is defined from the pivot.
How is GPE defined?
What is the equation for the gravitational attraction between two bodies?
the integral of the gravitational force between two points.
For a spherical body:
(GMm)/r
F = G (Mm)/r^2
What is the range of values that m_j can take?
magnitude <= j
2j + 1 states
What is the ideal gas equation?
What is the other key ideal gas property?
PV = NkT = nRT
E == E(T)
What is the boltzmann distribution?
p(state E) =~ e^(- E/ k T )
Remember to normalise with the partition function
What is the heat capacity of a solid (high T)?
3R
What is the half-life?
What about the lifetime?
halflife = ln(2) / lambda = lifetime * ln(2)
lifetime = 1/ lambda
What does adiabatic mean? What relation can we use for adiabatic expansion/compression?
Adiabatic: no heat transfer dQ = 0
P * V^gamma = constant
gamma = Cp / Cv
What is the first law of thermodynamics?
The change in internal energy = the heat supplied to the system + the work done on the system
How to work out the direction of cross products?
a x b
first finger - a
second finger - b
thumb points in direction of cross product
What is the approximate molar mass of air?
29g
What are the constituents of a K- meson?
ubar s
What are the constituents of a K0bar meson?
dbar s
If J = J_1 + J_2, what different j states can be formed?
j = j_1 + j_2 , … | j_1 - j_2|
What is Bragg’s law?
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]
What is the difference between an elastic and inelastic collision, and in which is there the maximal velocity transfer?
Elastic collision: momentum and KE conserved. (maximal velocity transfer)
Inelastic collision: momentum conserved
What are two expressions for current density?
j = sigma E = n q v
n is number density, v is drift velocity
sigma is electrical conductivity
What is fermat’s principle?
Light minimises the optical path (integral w.r.t. path of the refractive index)
What is a typical separation of particles in a gas, liquid, solid?
Gas ~ 10 angstrom
Liquid/solid ~ 1 angstrom
What are some suvat equations?
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.
What is the definition of entropy in statistical thermodynamics?
S = k ln(Omega)
where Omega is the #microstates corresponding to the macrostate
What is the magnitude equation?
What is the difference between flux density and luminosity?
What kind of scale is the magnitude scale?
m2 - m1 = 2.5 * log{10} ( F1 / F2 )
flux density = luminosity / (4 pi r^2)
magnitude is an inverse logarithmic scale
What is the equation for constructive thin-film interference?
(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]
What is the nonrelativistic equation for doppler shift?
(delta lambda) / lambda = v / c
(delta f) / f = v / c
What is the defining equation for refractive index?
n = c/v = lambda(vacuo) / lambda(medium)
frequency is constant
How are D and H defined?
D = e_r e_0 E B = mu_r mu_0 H
What is the 3D density of states in k-space, and how can the equivalent for energy-space be found?
g(k) = ( V k^2 / 2 pi^2 ) * g(s)
use E = (hbar^2 k^2) / (2m)
What is the reduced mass?
(Mm)/(M+m)
What is the zeeman splitting potential (strong field)?
delta E = 1/2 * mu_B * B * (m_l + 2m_s)
What is the eigenvalue of J^2 operator on an eigenstate?
What about the J_z operator?
hbar^2 j(j+1)
hbar m_j
What is the equation for distance to an object using parallax?
distance (parsecs) = 1 / the parallax angle (arcseconds)
What is the Taylor expansion equation?
–
What are the lorentz transforms for a frame S’ moving in the positive x direction, with frames co-linear at t=0?
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.
What is the acceleration of a point undergoing circular motion?
a = - v^2 / r = - omega^2 r
What is Malus’ law?
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.
What is the capacitance of a parallel-plate capacitor?
C = e_0 A/d
C = Q / V
How to “motivate” the schwarzchild radius?
Equate classical KE and GPE (GMm/r), set v = c.
2 G M / c^2
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?
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)
What is snell’s law?
n1 sin( theta1 ) = n2 sin( theta2 )
Angles are defined from the normal of the surface.
What is the range of visible wavelengths of light?
400 to 800 nm
What is brewster’s angle?
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.
