Kevin Zhou Physics Olympiad Handouts

Special Topics Review

1 Thermodynamics
[3] Problem 1. A photon of energy E bounces between two mirrors separated by a distance L.

(a) If the mirrors are slowly moved together to a distance L/2, find the final energy E by consid-
ering the blueshift the photon experiences at every collision.

(b) Check this result agrees with the adiabatic theorem.

(c) Using your result, infer the value of γ for a one-dimensional photon gas. Can you also infer
the value of γ for a d-dimensional photon gas?

[3] Problem 2. Let n be the local density of a gas of particles. If this density is nonuniform, it will
tend to be smoothed out by diffusion, which produces a particle current

J = −D ∇n

with units of particles per second per unit area. This is a continuity equation, as explained in T2,
and D is called the diffusion coefficient. In addition, we know from M7 and T1 that each particle
in such a gas will experience a drag force from the others. As a result, when a constant force F is
applied to one, its terminal velocity satisfies F = µv.

(a) Suppose the particles are placed inside a potential V (r). Write down the contribution to the
particle current due to the resulting force, neglecting diffusion.

(b) In thermal equilibrium, the particle current produced by the force, which pulls the particles
to lower potential, balances the particle current produced by diffusion, which spreads them
out. Assuming the temperature is T , find a relation between D and µ.

Remark
The result of problem 2 is called the Einstein relation, and is one of the four major
results Einstein derived in his “annus mirabilis” of 1905, the others being special rela-
tivity and E = mc2 , and explaining the photoelectric effect with photons. So why is
this result so important? It’s because at the time, it was not yet completely accepted
that matter was made of atoms. As you saw in T1, the size of a single atom often
drops out of kinetic theory calculations. At the time, many took this to mean that
atoms were a fictitious calculational tool, like how one does integrals by summing over
intervals of length ∆x, then gets a result independent of this fiducial length by taking ∆x → 0.

But doesn’t the number of atoms N appear all the time in basic thermodynamics, like the
ideal gas law? Yes, but always in the combination nR = N kB . Thus, information about
the size of a single atom is equivalent to information about kB . Einstein’s relation is useful
because it explicitly gives us kB in terms of the separately directly measurable quantities D,
µ, and T . It was one of the first unique, quantitative predictions of kinetic theory.

1
 Kevin Zhou Physics Olympiad Handouts

[4] Problem 3 (Physics Cup 2018). Estimate the mean free path of a heavy black sphere of mass m
and radius R in vacuum at temperature T . Here, we define the mean free path as the typical distance
it takes for the velocity vector of the sphere to turn by an angle π/2. Assume that kB T R ≳ ℏc.
(Hint: for a random walk where stepspof size a are taken per time τ , the average overall displacement
after time t ≫ τ is approximately a t/τ .)
[3] Problem 4. Consider a layer of the atmosphere with density ρ, pressure P , adiabatic index γ, and
density and pressure gradients dρ/dz and dP/dz. Suppose that a small parcel of air in this layer
acquires a small upward velocity. Under certain conditions, the parcel of air will begin oscillating
in height, performing simple harmonic motion. Neglecting drag and heat transfer between the
parcel of air and its surroundings, find the angular frequency ω of this motion. This is called the

01^‚
Brunt–Vaisala or buoyancy frequency.

01^‚
[3] Problem 5. USAPhO 2021, problem B2. A conceptual problem on methods of heat transfer.

[3] Problem 6. USAPhO 2019, problem B2. A useful problem for getting comfortable with

01^‚
numbers and estimates in astronomy.

01^‚
[3] Problem 7. USAPhO 2022, problem B1. A data analysis problem about a nonideal gas.

01hƒ
[3] Problem 8. USAPhO 2024, problem B2. Two exercises on the heat capacity of solids.

01hˆ
[3] Problem 9. INPhO 2019, problem 6. A thermodynamic cycle with a nonideal gas.

[5] Problem 10. IPhO 2011, problem 2. A problem on an electrified soap bubble, which combines
electrostatics, thermodynamics, and surface tension.

2 Relativity
[4] Problem 11. 01T†
IPhO 1998, problem 3. A great problem on a real controversy in physics, which

01T†
also gives you practice working with real data.

[4] Problem 12. EuPhO 2024, problem 2. A problem on visual perception in special relativity.
Note that in part (b), when the problem asks about what Alice “sees”, it means what she sees from
light reaching her eyes.
[4] Problem 13 (Physics Cup 2018). A spaceship travels with a constant proper acceleration g along
a straight line. At a certain moment, it launches two missiles in the direction of its motion, with
speeds v and 2v. Find the proper time interval in the spaceship between catching up to the first
and to the second missile.
[2] Problem 14. In classical physics, light waves do not interact with each other, but they can due to
quantum mechanical effects. Suppose two photons traveling in opposite directions scatter off each
other. Initially the photons have wavelengths λ1 and λ2 . One of the two outgoing photons exits at
an angle θ to the first incoming photon. Find its wavelength λ in terms of λ1 , λ2 , and θ.
[3] Problem 15 (MPPP 195). The pion π + is a subatomic particle with mass mπ . In one of its possible
decay modes, it decays into a positron e+ of mass me and an electron-neutrino νe of negligible mass.
What is the minimum speed of the pion if, following its decay, the positron and neutrino move at
right angles to each other? Numerically evaluate this speed in the limit me ≪ mπ .

2
 Kevin Zhou Physics Olympiad Handouts

[3] Problem 16. 01^‚ USAPhO 2023, problem B3. A nice, qualitative problem on supernova neutrinos.

[3] Problem 17 (Purcell 6.68). Consider two electrons moving side-by-side with parallel velocities,
with speed v and separation r. We wish to compute the three-force between the electrons.

(a) Compute this force by working in the electrons’ rest frame and Lorentz transforming back to
the lab frame.

(b) Compute this force by using the electric and magnetic fields of a moving charge, and verify
the answer agrees with that of part (a).

(c) What happens to the three-force as v → c?

3 Waves
[2] Problem 18 (OPhO 2024). A string of length L is attached at its endpoints to walls, with a
fixed tension. Initially, it is vibrating at its fundamental frequency with a small amplitude A. A
frictionless finger, initially at the right wall, slowly slides towards the left, flattening the oscillation
as it goes. When the vibrating part of the string has length L/2, find the final amplitude.

[3] Problem 19. EFPhO 2015, problem 2. A very nice combined interference and optics problem.

[3] Problem 20 (Crawford 4.15). Consider a jug with a large volume V , along with a thin neck of
length ℓ and area A. The lowest frequency standing wave can be excited by blowing across the neck.
Such a system is called a Helmholtz resonator. Naively, the corresponding wavelength would be
four times the length of the jug, but it is observed to be much larger. (Try it at home!)
Since the jug has a neck, the standing wave profile looks very different from a standard profile.
Most of the air motion is within the neck; the body of the jug serves as a large air reservoir that
acts as a spring pushing back against this motion. The frequency can be quite low, because this
reservoir is large.

(a) Show that if the air in the neck moves by a distance x, the restoring force is

γP0 A2 x
F =−
V
where P0 is the original pressure in the jug, and γ is the adiabatic index.

(b) Show that this mode has an angular frequency of

(c) See if this is roughly consistent with a real jug (e.g. a 1 liter soda bottle). Does the frequency
vary as you’d expect as you add water to the jug? If you feel musically inclined, can you find
how to excite higher frequencies?

3
 Kevin Zhou Physics Olympiad Handouts

Remark
The results of problem 20 can also be used to describe window buffeting, the annoying
“whuppa, whuppa, whuppa” sound you get when you slightly open one window of a rapidly
moving car. Suppose the opening has√area A and the car has volume V . Then the incoming
air moves quickly up to a depth ℓ ∼ A in your car, and this region functions as the “neck”.
The resonant frequency is therefore f ∼ vA1/4 /V 1/2 . You can stop the buffeting by opening
the window more, increasing A so the frequency is too high to be efficiently driven by the air.

Remark
The udu is a Nigerian musical instrument which can be modeled as a Helmholtz resonator
with two holes. The holes have different sizes, so you get a frequency f1 if you cover one
hole, and f2 if you cover the other.

You can get a third frequency by hitting the instrument with both holes open. To find that
third frequency, note that a Helmholtz resonator is like a spring-mass system, where the
“spring” is the air in the instrument, and the “mass” mi is the air near hole i, which moves
back and forth. When both holes are open, we effectively have a spring with masses attached
at both ends, which oscillates according to the reduced mass 1/µ = 1/m1 + 1/m2 . Since
√ p
fi ∝ 1/ mi , the third frequency is f1 + f22 .
2

[3] Problem 21. 01W USAPhO 2008, problem A4. A neat Doppler shift problem that also tests your

01mƒ
data analysis skills.

[4] Problem 22. INPhO 2020, problem 4. A nice and tricky problem on the two-dimensional
Doppler effect. (Incidentally, this problem was also on the 2013 Russian Physics Olympiad, but I’m
linking to this one because it’s in English.)

[3] Problem 23. 01^‚USAPhO 2022, problem A3. A great problem on the rainbow caustic. For some
brilliant rainbow graphics, see here and here.

4 Modern
[3] Problem 24. Consider a typical small LC circuit, with L = 10−7 H and C = 10−11 F.

(a) If the circuit is at zero temperature, estimate the voltage uncertainty across the capacitor.

(b) Do the same if the circuit is at room temperature.

Today, some of the leading quantum computing hardware is based on such circuits. Because thermal
fluctuations would ruin the desired quantum mechanical effects, the circuit must be cooled so that
kB T ≪ ℏω, which corresponds to a small fraction of a degree. When you see pictures of quantum
computers, most of what you’re looking at is the fridge!

[2] Problem 25. INPhO 2013, problem 2. A short problem on X-ray diffraction.

To finish up, here are three neat questions that each cover a broad range of topics.

4
 Kevin Zhou Physics Olympiad Handouts

[5] Problem 26. 01hˆ IPhO 2009, problem 3. Estimating the size of a star from scratch.

[5] Problem 27. 01hˆ IPhO 2021, problem 3. Molecular fluorescence and optical lattices.

[5] Problem 28. 01hˆ USAPhO+ 2021, problem 2. A full analysis of a real dark matter detector.

Remark
Problem 26 estimates the size of a star using a rough treatment of the nuclear physics
we know applies at its core. When you think about it, it’s amazing that this is possible
at all. There is essentially no way to directly probe what’s going on inside any star. We
can basically only measure the size of the star, its rough age, and the temperature of its
surface. But the models work! This was the result of generations of painstaking effort,
which stimulated progress in both nuclear and particle physics. To learn the history of solar
modeling, see this article for the early days and this article for the eventual triumph. For
more estimates of the sizes of astronomical objects, see this article.

It is even more remarkable when you realize that this sort of story applies to essentially all
of the dozens of known types of astronomical objects. For most of these, astronomers can
directly measure only the electromagnetic spectrum and its variation over time, but that little
is enough to develop and test detailed physical models. Most people think astronomy is very
concrete, because they constantly see stars in the night sky, and flashy “artist’s impression”
graphics on the internet. But the field is actually extremely theoretical, with every insight
won through extensive calculation.