Theoretical problems: solutions. Language: English

IPhO 2022

General rules
Rule 1: Propagating errors are not punished except(a) those cases when the (possibly intermediate) result isclearly wrong and hence, the contestant has another opportunity to figure out that something must have gonewrong, and could start searching for the mistakes. Examples (not limited to): dimensionally wrong answer;
obviously too large or too small numerical value; wrong
sign of the effect;(b) for the calculation of numerical answers: marks(usually 0.1 pts) for the numerical values are given only
when the underlying formula is correct.
Rule 2: if a contestant writes down a final answer (formula) of a task, or an intermediate result needed to derive the final result, the marks are not given even if theformula is completely correct, unless it is judged thatthat particular formula can be obtained from the basic
laws in a simple-enough-way so that could have been
done purely in mind, without using paper.

force on the ring is F = Bp Iπd, where Bp is the magnetic field component lying in the plane of the ring at
its line element. From the given formula of dipole field,
3µ0 m sin θ cos θµ0 m. Sub|Bp | = 4πr
3 (2 cos θ sin θ + sin θ cos θ) =4πr 3dstituting tan θ = 2L with small angle approximation for
20m
sin θ ≈ θ and cos θ ≈ 1, one gets F = 3µ2πL4 .

a) m = π4 d2 hJb) I = Jhsin θ cos θc) |Bp | = 3µ0 m4πr
33µ0 m2d) F1 = 2πL4e) small angle approx.
or correct limiting behavior
Answer: 0.21 mN

0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts

T1: Permanent magnets
Part A: Interaction of two magnets.
(A.1) Solution 1.
For all questions, note that spurious rounding (e.g.
rounding 2.4kA to 2kA) will incur a penalty of 0.1 points
lost per incident. Although such rounding does technically adhere to rules about significant figures, IPhO is notmeant to test one’s understanding of significant figures.
Such rounding is both unreasonable and makes checking solutions much more difficult. Since the distance between the magnets is big as compared to their size, we
can approximate each of them as a dipole of magnitudem=

π 2
d hJ = 0.75 A m2 .
4

Since the two dipoles are parallel to each other and to
the line connecting them, the interaction energy of oneof the magnets with the field of the other magnet is
2

µ0 m
⃗ ·m
W = −B⃗ =−,2πL3and by taking a derivative we obtain
F1 =dW
3µ0 m2≈ 0.21 mN.
=
dL
2πL4

a) m = π4 d2 hJ⃗ ·mb) W = −B⃗c) F1 = dWdL 20md) F1 = 3µ2πL4Answer: 0.21 mN

0.2 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts

Solution 2. The force between the magnets is foundby converting one cylinder into a superconducting ring.
The current in the ring is I = m
A = Jh. Consider asmall line element dl on the ring with a radial vector
joining from the dipole to the line element. The dipole
moment is decomposed into radial and tangential component. By rotational symmetry along the co-axis, the(A.2) In the case of a homogeneous magnetization, themolecular currents in the bulk of the material cancel out,
leaving only a surface current at the surfaces which are
not perpendicular to the magnetization vector. Hence,
there is a surface current on the side surfaces of thecylinder. As the height of the surface is much smaller
than the radius, these currents can be approximated as aring current I; the dipole moment πd2 I/4 of the ring current must be equal to the total dipole moment πd2 hJ/4of the magnet, hence I = Jh ≈ 2.4 kA.
a) current around edge of magnetc) m = IAd) I = Jh
Answer: 2.4 kA

0.1 pts
0.1 pts
0.1 pts
0.1 pts

Remark: Having established the analogy to an edge
current, one could instead find I by evaluating the magnetic field at some point (e.g. far above the current loop)
and matching to the given dipole result. Biot-Savart at apoint on the ring axis at some distance z ≫ d above theloop givesµ0
B = B ẑ =4π

Z2πI(d/2)dθ d/2µ0 Id2
ẑ.
ẑ
=
z3
z
8z 3

(1)

0

Matching this to the dipole result from the previous partgives I = Jh as above.
(A.3) Since the distance between the magnets is now significantly smaller than their diameter, the force can beapproximately found as the force between two straightcurrents I of length πd at distance L:
F2 =µ0 I 2 dµ0 I 2πd =≈ 14 N.
2πL
2L

Theoretical problems: solutions. Language: English

IPhO 2022a) Consider as straight currentsµ0 Ib)B = 2πLc) F2 = πdIBI2dd) F2 = µ02L
Answer: 14 N

0.3 pts
0.3 pts
0.2 pts
0.1 pts
0.1 pts(A.4) The chain will most likely break below the topmost magnet because then the magnetic pull between
the magnets needs to compensate the largest possible
weight. Let the number of magnets be N + 1, and themass of a single magnet M = π6 ρδ 3 ≈ 0.5 g; then theweight of the magnets F = M N g is balanced by the magnetic force
3µ0 m2 X 1µ0 m2 π 3=,442πδ n=1 n
60δ 4
N

F =

where m = π6 Jδ 3 ≈ 78 mA m2 and we have assumed thatN ≫ 1 so that we can assume in the sum N = ∞. From
the force balance we obtainN=µ 0 m2 π 3≈ 1320;60M gδ 4

hence, the total length of the chain is N δ = 6.6 m. Notethat N = 1320 is indeed much bigger than 1.
a) It will break at the topb)M = π6 ρδ 3c) F = M N g
2 PN
1
0m
d) F = 3µn=1 n42πδ 4π3e) m = 6 Jδ
2 30m πf) l = µ60Mgδ 3
Answer: 6.6m

0.1 pts
0.1 pts
0.2 pts
0.2 pts
0.2 pts
0.1 pts
0.1 pts

Remark: if the sum is substituted with a finite sum as an
approximation, with two or three terms in it, full marks
are given. If only one term is kept, subtract 0.1 from d) orf). Remark 2: It’s possible to get a range of final answers
depending on the approximations used for g, mass, magnetic moment, etc. Answers that round to 1300 balls
should definitely not be penalized, which corresponds
to a distance range of 6.25 - 6.75m. 1260 balls (6.3m) is
what you get with g=10 and mass = 0.5g; 1320 balls (6.6m)
is what you get with g=9.8 and a mass of 0.49g (or, without rounding the mass and magnetic moment and cancelling out the volume).
(A.5) Solution 1. Each of the balls creates magnetic fieldof a dipole m; the magnetic dipole creates the same fieldwich would be created by two magnetic charges, equalby modulus to q and of opposite sign, at a distance s =m/q, assuming that this distance s is much smaller thanthe distance from the dipole to the observation point.
Here it is convenient to select s = δ (hence q = m/δ)
because in that case almost all the positive and negative magnetic charges overlap and cancel out each other.
The only ones which will not cancel out are the magnetic
charges at the chain’s endpoints. One of these charges isvery far so that the field at P is the field of a magnetic
charge at O:B=µ0 m
Jµ0 δ 2
µ0 q
=
=
.
2
2
4πr
4πδr
24r2

a) Idea of magnetic chargesb)q = m/δµ0 qc) B = 4πr2Jµ0 δ 2d) B = 24r2

0.4 pts
0.4 pts
0.4 pts
0.3 pts

The same scheme applies for solutions which work withelectrical charges, with a proportionality constant relating that field to the magnetic field of magnetic dipoles.
Then, the sub-score a) is given for the idea of calculating
the field of electrical dipoles (0.2 pts), with a correct proportionality factor between the two fields, k = B/E =µ0 ε0 = c−2 (0.2 pts).
Solution 2. It is clear that from distances larger thanthe diameter of a magnet, the shape of the magnets
doesn’t matter; what matters is only the total dipole moment as this is what defines the magnitude of the fieldat large distances. So, we can substitute the balls withcylinders of equal volume. Now, let us require the heightof these cylinders to be δ; then the neighbouring cylinders in the chain will be touching each other. As a result, instead of the chain of balls, we have a long homogeneous cylinder. Equal volume means that the crosssectional area of these cylinders A = π6 δ 2 . We know
from task A.2 that such a cylinder can be considered as along solenoid carrying surface density of bound currents
equal to J. So, the magnetic field inside it B0 = µ0 J, andtherefore, it carries magnetic flux Φ = B0 A = π6 δ 2 µ0 J.
We know that inside the solenoid, magnetic field is constant, and outside, the field is zero. However, this isvalid only until the endpoints of the solenoid are far. All
that flux is released near each of the endpoints of thesolenoid. The released flux needs to satisfy the Maxwell
equations: the B-field needs to have no sources and bepotential. We know that the only solution in such a case
⃗ = f (r)r̂, where r denotes
is a central isotropic field B
the distance from the endpoint and r̂ — the corresponding unit vector. From the Gauss law we conclude that20δ
4πr2 f (r) = ´Φ = π6 δ 2 µ0 J, hence B = Jµ24r 2 .
a) Idea of substituting spheres with cylindersb)A = π6 δ 2c) Φ = π6 δ 2 µ0 Jd) B = Φ/4πr220δe) B = Jµ24r 20.4 pts
0.2 pts
0.4 pts
0.4 pts
0.1 pts

Remark: for part a, give only 0.1 points if students makethe cylinder replacement but then fail to make any real
progress using it.
Solution 3. This solution follows the solution 2 up to
the point where we have a solenoid with surface current density J. After that we observe that at any pointin space, the axial component of the magnetic field is
B = µ0 JΩ,
4π

where Ω denotes the solid angle under which we cansee the interior surface of the solenoid, minus the solid
angle under which we can see the outer surface. Thiscan be derived easily from the Biot-Savart law: dBz =µ0⃗⃗
4πr 2 jdzdl × r̂ · ẑ, where hats denotes unit vectors, dl —an infinitesimal vector parallel to the surface current,

IPhO 2022

Theoretical problems: solutions. Language: English

and ⃗r — a vector pointing from the observation point
to a point on the solenoid. This can be rewritten as
µ0
µ0
⃗
⃗
dBz = 4πrz × d⃗l· r̂ = 4πr
2 jd⃗
2 JdA · r̂, where dA denotes thearea of a surface element on the solenoid. To complete
our proof, it suffices to notice that ⃗dA · r̂ is the apparent
area of the surface element, dΩ = ⃗dA · r̂/r2 .

Integration for Bz :
!
Z
Z
2
1
3r2 sin θµ0 m ∞
dz ′3 2 −Bz = dBz =
4πδ 0
r
r′2
Z ∞
h
i
µ0 m
=
du 2(u2 + 1)−3/2 − 3(u2 + 1)−5/22
4πδr2 sin θ − cot θ∞
2u2u3 + 3u
µ0 m
√
=
−
2
Now, at the point P , the outside and inside contribuu2 + 1 (u2 + 1)3/2 − cot θ
4πδr2 sin θ
tions to Ω cancel out everywhere except for the circuµ0 m cos θlar opening of the solenoid. Thus, Ω = A cos θ/r2 so
=−
2
4πδr2
0δ
that Bz = Jµ24r 2 cos θ. Finally, we can use the Gauss
law to obtain expression for the radial component BR Integration for BR :Z
(with R denoting the radius in cylindrical coordinates)
BR = dBRof the magnetic field. Someone not familiar with vecrZ r cos θ
tor calculus can calculate the magnetic flux Φc through
3µ0 m
1 r
r2
2
a circle of radius R0 = r sin θ. Then, the cylindri=dz ′3 · ′ sin θ 1 − ′2 sin θ
4πδ
r
r
r
′
−2 ′
′
0
cal coordinate R = z tan θ so that dR =R z cos θ dθ ,!
r
Z ∞2πRBz dR =and r12 = cos2 θ′ /z 2 . Therefore Φc =
r2
1 r
2
2
2
πJµ0 δ
−dz ′3 · ′ sin θ 1 − ′2 sin θ0δ
cos θ′ ´dθ′ = πJµsin θ. We can see that this der
r
r
12
12
r cos θpends only the spherical coordinate θ; by considering
Z ∞
udu
3µ0 m
conical frusta with circular faces having the same polar=−2
2 + 1)5/22(u
4πδr sin θ cot θangle θ we can easily conclude that the magnetic field
Z ∞
2
0δ
3µ0 m
must be radial, i.e. B = Bz / cos θ = Jµ
.
24r 2
=−
dv(v + 1)−5/2 (v = u2 )2228πδr sin θ cot θµ0 m sin θ=−4πδr2a) Idea of substituting spheres with cylindersb)A = π6 δ 2
20δc) Bz = Jµ24r 2 cos θd) B = Bz / cos θ20δe) B = Jµ24r 20.4 pts
0.2 pts
0.4 pts
0.4 pts
0.1 ptsa) writing correctly Bszb) writing correctly BRzc) dm = mdz/δ20δc) Bz = Jµ24r 2 2cos θ0δd) BR = Jµ24r 2 sin θ
Jµ0 δ 2e) B = 24r2

0.2 pts
0.2 pts
0.2 pts
0.4 pts
0.4 pts
0.1 pts

Remarks: for c) and d), a partial credit of 0.1 pts canbe given for each of these integrals if the initial integral
Solution 4. Finally, the solution could be obtained the- is written correctly, but the calculation of the integral
oretically also by summing over all the fields of individ- is not performed or there are many mistakes. If only
ual magnets. However, this is mathematically very de- few mistakes were made, subtract 0.1 for each mistakemanding, therefore full solution is not provided here. made. If initial integral is written incorrectly, no points
The first steps are as follows. (i) Writing the contribu- are given. Points for e) are given only if the final answer
tion Bsz and BsR of a single magnet at distance z from is completely correct.
the point O to the axial and radial (in cylindrical coordi- Another remark: in the integration of B , if the changeRnates) components of the magnetic field; (ii) going from of sign (of the cosine) is ignored, the correct answer
summation of individual contributions to integration by could still be obtained (because the extra parts cancel
assuming linear density of dipoles ρm = m/δ so that out), but the derivation would technically be wrong.
dm = mdz/δ; performing integration over z to find the
Solution 5. It’s possible to perform the direct integrafield components.
tion of the previous solution more easily using angular
The mathematical derivation: A dipole at position z
variables in place of z. Let s = r sin θ be the distance of
dm = mδ dz generates a magnetic field (in Cartesian co- closest approach of the line to P for convenience and ϕordinates):
be the angle from a point on the line to P (such that ϕ = θat the end near P , ϕ ≈ π at the other end). Then the addi
tional magnetic field from a small component given byµ0 dm 2 ′dBz = dBr′ cos θ − dBθ′ sin θ =
2
−
3
sin
θ
dϕ is
4πr′3
3
3µ0 dmµ0 sin ϕdBR = dBr′ sin θ + dBθ′ cos θ =sin θ′ cos θ′(2dm∥ − dm⊥ )
dB
=
′3
4πr
4πs3
3
µ0 sin ϕ
dm×
=
√
4πs3
r
′
′
2
2
Where r = r + z − 2rz cos θ and sin θ = r′ sin θ are(2 cos ϕ(cos ϕẑ − sin ϕr̂) − sin ϕ(sin ϕẑ + cos ϕr̂))
coordinates relative to the dipole dm. In order to sim3cos θ
plify the integration, do substitution: u = z−rµ0 sin ϕ dm
r sin θ , then√=dz((3 cos2 ϕ − 1)ẑ − cos ϕ sin ϕr̂).
4πs3 dzr′ = r sin θ u2 + 1; dz = r sin θdu.

Theoretical problems: solutions. Language: English

IPhO 2022

Since s = −z tan ϕ and dm = mdz/δ, we have dm/dϕ =2sm/(δ sin ϕ). Then
dB =

µ0 m
2
((3 cos2 ϕ − 1)ẑ sin ϕdϕ − sin ϕr̂ cos ϕdϕ)4πs2 δ

and thusµ0 m
B=
4πs2 δ

ϕ=πZ(−(3 cos2 ϕ − 1)ẑd cos ϕ − sin ϕr̂d sin ϕ)
2

ϕ=θ



π
µ0 m
π
3
3
=−
cos ϕ − cos ϕ ϕ=θ ẑ + sin ϕr̂
4πs2 δ
ϕ=θ
µ0 m
2
3
=
(− cos θ sin θẑ + sin θr̂).
4πs2 δ
Putting back in our expression for s, we haveB=

µ0 m(− cos θẑ + sin θr̂)
4πr2 δ

(2)

which is the desired result.
a) writing correctly dB/dϕ2b) dm/dϕ = sm/(δ sin ϕ)c) |Bz | = µ0 m/(4πr2 δ)d) B̂ = − cos θẑ + sin θr̂

0.5 pts
0.2 pts
0.4 pts
0.4 pts

Remark: for d, note that the coordinate system wasn’t
specified in the problem, so check what the student is using; the point is to get the “radially outward” (or inward)
idea.

Part B: Interaction of magnets with ferromagnetic materials.

2

3

1

2

(B.1) Due to the boundary condition at the surface of theferromagnet, the field lines must enter the plates almost
perpendicularly. Indeed, as it follows from the Ampère’s⃗ is continucircutal law, the tangential component of B/µous at the surface of a ferromagnet; similarly, the Gausslaw for the magnetic field implies that the normal component of the B-field is continuous. From these two facts,
one can derive the “refraction law” for the field lines,
tan α = µ tan β, where α and β are the angles between
the tangents of a field line and the surface normal, insideand outside of the ferromagnetic, respectively. From thefact that µ ≫ 1 we can deduce that as long as α is not
small, β ≈ 0. Those field lines which enter the plate
must exit it somewhere, this happens somewhere farther away from the magnet, see the sketch below.

a) Field line 1 correctb) Field line 2 correctc) Field line 3 correct

0.2 pts
0.4 pts
0.4 pts

Remarks:i) Subtract 0.1 both from b) and c) if the field line does
not enter the plate perpendicularly;ii) Subtract 0.1 both from b) and c) if the field line does
not refract correctly;iii) Subtract 0.1 from (b) if its segment rightwards of themagnet is not shown (note that in the student answer
sheet, the magnet is to the right of the plate, not on topof it as shown in the solution);iv) Subtract 0.1 both from a) and c) if the field line does
not form a closed loop;v) Subtract 0.1 from a) if the line touches or enters theplate; 0.1 from b) if the line exits the plate; 0.1 from c) if
the line reaches the other side of the plate.
(B.2) The problem can be solved by introducing an image magnet — a mirror reflection of the real magnet withrespect to the surface of the plate, with the dipole moment being both reflected and flipped. With this image magnet, the boundary condition above the plate issatisfied: the field lines enter the plate perpendicularly.
Hence, the force and torque exerted to the real magnet
are equal to the force and torque exerted by the image
magnet. The equilibrium is achieved when the dipole isparallel to the field created by the image magnet whichis the case when the dipole moment is perpendicular tothe plate. Hence, leftmost boxes of the first and second
row need to be marked with a tick. The force is almost
the same as what was already found in part A(d), withthe only difference that now there is only the first termin the sum:
3µ0 m2F == 5.9 N.
2πδ 4a) Idea of magnetic image (even if J⃗ not flipped)b) Correct direction of the image J⃗
20mc) F = 3µ2πδ 4d) F = 5.9 Ne) each correct tickf) each incorrect tick

0.3 pts
0.2 pts
0.2 pts
0.1 pts
0.1 pts-0.1 pts

Remark: if e) + f) adds up to a negative number, replace
the total score for those two parts by 0.
(B.3) Solution 1. As explained above, the magnetic fieldlines are perpendicular to the surface of the ferromagnetic plate. Since the gap is narrow as compared to its
width, the field lines are inside the gap almost straight.
Due to the Ampère’s circulation theorem it also meansthat the field in the gap is homogeneous. Due to the
Ampère’s circulation theorem, field outside the gap vanishes as the gap’s width tends to 0, so in the limit all flux
through the permanent magnet wraps around throughthe gap; see the sketch of magnetic field lines. Now, let
us recall that the disc magnet is equivalent to a surface
current of density J along the curved surface of the disc.
Hence we can write the circulation theorem along theloop defined by one of the field lines shown in the figure:I⃗ · d⃗r ≈ (B1 + B2 )h/µ0 ,I= H
where B1 and B2 denote the flux density inside the permanent magnet and outside the magnet (but still inside
the slit), respectively. Here we have neglected the contribution of the magnetic field inside the ferromagnetic

Theoretical problems: solutions. Language: English

IPhO 2022

plate to the integral because µ is very big. Due to theGauss law, π4 d2 B1 = π4 (D2 − d2 )B2 ; with D = 2d this yields
B1 = 3B2 . Thus, B2 = Iµ0 /4h = Jµ0 /4 = 0.375 T and
B1 = 1.125 T. In order to find the force exerted to oneof the ferromagnetic plates, we can notice that the force
does not depend on what is creating the magnetic field
and, hence, we can substitute the disc magnet with thecurrent I in a superconducting ring. Next we apply thevirtual displacement method and increase the distance
between the plates by dx. In the case of a superconducting ring, the magnetic flux through the ring is conserved,
and therefore, the magnetic field strength inside the gap
will remain unchanged during the virtual displacement.
With all this information we are ready to calculate thechange of the magnetic field energy. The magnetic fieldenergy inside the ferromagnet can be neglected because
its density is ca µ times smaller than inside the gap. So,
the energy is changed only because the volume of the
gap is changed:


π 2 2
3π 2 222
2dW =[d B1 + (D − d )B2 ]dx =B d dx8µ0
2µ0 2which means that the forceF =dW
3π 2 23π 2=B d =J µ0 d2 ≈ 210 N.
dx
2µ0 2
32

manent magnet in the magnetic field, ´
3Iµ0π= −2Wf ,Wm = −mB1 = − d2 I ·44hhence the total energy W = −Wf . Now we can find force
dWf
as F = − dW
dh = dh , yielding the same result as before.
Notice that if we didn’t take into account the energy ofthe dipole then we would have obtained the correct answer by modulus, but with a wrong sign — we would
have had repulsion instead of attraction of the plates.
⃗ in the slit is homogeneousa) B⃗ in the permanent magnet is homog.
b) B⃗c) B in slit and in perm. magn. is normale) I = (B1 + B2 )h/µ0f) π4 d2 B1 = π4 (D2 − d2 )B2
g) B2 = Iµ0 /4hh) B1 = 3Iµ0 /4hi) Wf = 8µπ0 [d2 B12 + (D2 − d2 )B22 ]hj) Wm = −Wfk) F = dW
dh
2
2
l) 3π
32 J µ0 dm) F ≈ 210 N.

0.2 pts
0.2 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.2 pts
0.1 pts
0.1 pts
0.1 pts

Remark: those who do not take into account the energyof the dipole will be given zero points for j), k), l), and m).
⃗ in the slit is homogeneousa) B
0.2 pts
Solution 3. There is a more intuitive way of finding⃗ in the permanent magnet is homog. 0.2 ptsb) B
the field distribution. Using carefully the analogy be⃗ in slit and in perm. magn. is normal 0.1 pts
tween electric and magnetic dipole fields, one could conc) B
vert the problem into a permanent electric polarizatione) I = (B1 + B2 )h/µ0
0.1 pts
inserted between two conducting plates. From similarif) π4 d2 B1 = π4 (D2 − d2 )B2
0.1 pts
ties among Maxwell equations, it could be observed thatg) B2 = Iµ0 /4h
0.1 pts
E ≈ B, D ≈ H and P ≈ M , with some prefactors involvh) B1 = 3Iµ0 /4h
0.1 pts
ing permeabilities and permittivities. Consider puttingi) dW = 8µπ0 [d2 B12 + (D2 − d2 )B22 ]dx
0.3 pts
the smaller capacitor inside the conductor plates, thej) F = dW
0.1 ptsdx
3π 2
2charge would induce opposite charge that makes fieldk) 32 J µ0 d
0.1 pts
lines perpendicular to the conductor. In addition, therel) F ≈ 210 N.
0.1 pts
should not be net charge on the metal plates upon insertion of the smaller cylinder. Hence, there is again an
uniform charge density of opposite charge on the metal
plate spreading over the larger region. Effectively, for E
field, it is equivalent to spreading the original charge on
smaller plate onto the larger plate. Because the radii has
Qratio of 2, E = 4SQ0 ϵ0 , D2 = 4S
, and D1 = SQ0 (1 − 1/4) =0
3D2 . This agrees with B2 , B1 in previous solutions. The
rest easily follows. (There are confusions about B or H
Solution 2. The second solution follows mostly the first but most are due to the definition of polarization charge
one, and deviates only after the fields B1 and B2 have or current being considered free or not, a self-consistent
been found. Now we do not introduce the fictitious su- derivation would be sufficient.The close-to-centre part ofperconducting loop, and instead calculate carefully all the field of electric and magnetic dipole is opposite andthe changes to the magnetic field energy during virtual one should be careful about this effect inside polarisadisplacements. Now the current around the perimeter of tion when utilizing the analogy.)
the permanent magnet is fixed to I as its magnetisation
is assumed to be constant. We can still use the previous
expressions for the magnetic field energy if we consider
the distance h between the plates to be a variable:
Wf =

3Iµ0
3µ0 πd2 I 2
πd2 h 2[B1 +3B22 ], B1 = 3B2 =⇒ Wf =
.
8µ0
4h
32h

In addition to the change of the magnetic field energy,
we also need to take into account the energy of the per-

IPhO 2022

Theoretical problems: solutions. Language: Englisha) correct analogy argumentsb) correct charge distributionsc) E is uniforme) correct D expressionsf) correct conversion factorg) B2 = Iµ0 /4hh) B1 = 3Iµ0 /4hi) Wf = 8µπ0 [d2 B12 + (D2 − d2 )B22 ]hj) Wm = −Wfk) F = dW
dh
2
2
l) 3π
32 J µ0 dm) F ≈ 210 N.

0.2 pts
0.2 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.2 pts
0.1 pts
0.1 pts
0.1 pts

Solution 4. (Incorrect) A solution that has been submitted by a few students uses an infinite number of imagesof the permanent magnet. This forms an infinite rod,which they assume gives the same magnetic field as anormal magnetised rod would, 0 everywhere outside it.
However, since the plates are finite, the magnetic fieldoutside would actually be non-zero, and would need tobe calculated according to Solution 1. In this case, only
marks corresponding to a), b) and c) in the scheme of Solution 1 should be awarded, i.e. 0.5 marks. If someone
doesn’t assume the field outside to be 0, give marks forthe subsequent calculations according to solution 1.

always pointing in the direction of x̂ which ensures therotational stability of the magnet. Attraction force between two neighbouring rows is contributed only by thevertical nearest-neighbour pairs of balls, so we can justcalculate only the interaction force between two suchmagnets. If two such balls were to be at distance y, the20m
interaction energy would be W = ± µ4πy3 so that the y3µ0 m
directional force Fy = dW
dy = ∓ 4πy 4 . This means that thetwo balls attract if they are antiparallel and repel otherwise. This brings us to the conclusion that the order must
be antiferromagnetic, shown below in the sketch.
2

The work needed to pull out one of the magnets is easily found as its interaction energy with the four near⃗ · m,
est neighbours, with minus sign, i.e. W = B⃗ where
3µ0 m
3µ0 m2⃗⃗⃗
B = 2B1 + 2B2 = 2πδ3 x̂ so that W = 2πδ3 = 29 mJ.
a) Fig: left and right parallel magnets attractb) Fig: top and bottom antipar. magn. attract⃗ from the 4 neighbours ∥ mc) B⃗ ⇒ no torqued) correctly marked 12 arrowse) antiferromagnetic⃗ ·mf) W = B⃗
3µ0 m2g) W = 2πδ3h) W = 29mJ

0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts

Remarks: no marks for d) if any of the magnets has aPart C: Model of ferromagnetic and anti- wrong direction or has no arrow. No marks for e) if theferromagnetic materials.
score for d) is zero.
Solution 2, It appears that there is another stable con(C.1) Solution 1, Since the task is about finding only one
figuration, see figure below
configuration of dipoles, we can just try looking for configurations satisfying the requirements. The simplest approach is to start construction with the chain of magnets
described in part A.4: if all the dipoles are directed parallel to each other and parallel to the chain, the system isobviously in equilibrium. Now, two such chains can beparallel to each other, and they can be also antiparallel.
In both cases, each of the balls is in a stable equilibriumin terms of rotations. Indeed, each of the balls from the⃗ 1 = x̂ µ0 m3 ,left and from the right contribute the field B
2πδ
while each of the balls from above and below contribute⃗⃗2 = ±1B
B
2 1 , where x̂ denotes a horizontal unit vector; ’+´corresponds to antiparallel rows, and ’−´ — to parallelrows. Since B2 < B1 , the sum of the four contributions is

Theoretical problems: solutions. Language: English

IPhO 2022(C.2) Now we need to repeat the steps done for the previous question, with the only difference in the mutual
placement of the magnets. Also, each of the magnetsof the top row interacts now with two magnets of thebottom row with the three magnets forming a equilateral triangle. Since we’ll be going to use virtual displacement method, we consider the interaction of three magnets forming an isosceles triangle as shown in the figure;while the base of the triangle remains fixed during virtual displacements, the length of the sides l will change.

B
A

C

E
D

In order to show that this configuration is stable, too,
let us find the direction of the magnetic field at the position of one of the balls, the ball marked with A in the
figure, due to its four neighbours. Using the formula forthe magnetic field of a dipole, we can see that the balls B
and D create both field b(x̂ + 2ŷ), where x̂ and ŷ are horizontal and vertical unit vectors. Meanwhile, both C andE create field b(2x̂ + ŷ) so that the total field is 6b(x̂ + ŷ);this is parallel to the dipole moment of the ball A whichmeans that no torque is exerted on it. What is left to do
is to calculate the interaction force between two neighbouring balls, e.g. A and B. One way to do it is to decompose the both dipoles into vertical and horizontal components: √m⃗ A = m0 (x̂ + ŷ and m⃗ B = m0 (−x̂ + ŷ, wherem0 = m/ 2. One can easily see that the pair of dipolesm0 x̂ and −m0 x̂ attract, and the same applies to the pairm0 ŷ and −m0 ŷ. It is also easy to see that there is no horizontal component for the interaction force between theremaining pairs, m0 x̂ with m0 ŷ and m0 ŷ and −m0 x̂. A
little more efforts are needed to see that the horizontal
component of the interaction force is also zero. To thatend one can calculate first the torque TAB exerted bydipole A to B with respect to the centre of the ball B, andthe torque TBA exerted by B to A with respect to the centre of the ball A; one can easily see from symmetry that
TAB = −TBA . Due to Newton’s third law, with respect tothe centre of the ball A, the sum of torques exerted by
B to A and by A to B must be zero; it can be expressed
as TAB + TBA + Fx δ = 0, where Fx denotes the horizontal component of the force exerted by A to B. From thisequality we can conclude that Fx = 0. So we found thateach of the neighbouring balls attract each other, hence
the whole configuration is stable.
a) Showing: neighbouring magnets attract⃗ from the 4 neighbours ∥ mb) B⃗ ⇒ no torquec) correctly marked 12 arrowsd) antiferromagnetic⃗ ·me) W = B⃗
3µ0 m2f) W = 2πδ3g) W = 29mJ

0.2 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts

Remark: no marks for c) if any of the magnets has awrong direction or has no arrow. No marks for d) if thescore for c) is zero.

First we need an expression of the magnetic field causedby the two bottom magnets at the centre of the topmost
magnet. Due to symmetry, this field must be horizontal;
we can use the formula provided in the problem text forfinding it. The dipole moment of the left-bottom magnet
needs to be divided into components parallel and perpendicular to the radius vector drawn from its centre
to the centre of the topmost magnet, m∥ = m cos α andm⊥ = m sin α. Hence, we can express the resultant xcomponent of the magnetic field asµ0 m⃗⃗ 3x = µ0 (2m⃗ ∥ cos α − m⃗ ⊥ sin α) =(3 cos2 α − 1).
B
4πl3
4πl3
The magnetic field due to both magnets is therefore
2B3x x̂.
As the first thing, we can now analyse the stability of amagnet with respect to rotations. As before, we assume
that the magnets in one single row are parallel to eachother, and the magnets at the two neighbouring rows
are either parallel or antiparallel to each other. In either
case, the rows at the top and at the bottom from a given
magnet are parallel to each other; let them be oriented
along x̂. Then, each row contributes 2B3x x̂ to the total
field at the position of our magnet. The total field has⃗ 4x = ± µ0 3 from the left and rightalso contributions B
2πδ
magnets; here ‘+’ corresponds to the ferromagnetic order, and ‘−’ — to the antiferromagnetic order. Keeping
in mind that l = δ and cos α = 12 the total field is⃗ 5 = 4B
⃗ 3x + 2B⃗ 4x = µ0 m − 1 ± 2 x̂.
B
2πδ 3
2
This is parallel to the given magnetic dipole for both ‘+’and ‘−’, which ensures stability in any case.
With m⃗ = ±x̂m and y denoting the height of the isosceles triangle, the vertical component of the interaction
force of a magnet with a magnet in the bottom row canbe found as
dl 3µ0 m2
dl d µ0 m2 3δ 2
d ⃗
−
1
=∓
;
B3 · m
⃗ =±
F5y =
3
2
dy
dy dl 4πl4l
dy 16πδ 3δhere we have used cos α = 2land upon taking derivative, substituted l = δ. For this force to be attractive, weTheoretical problems: solutions. Language: English

IPhO 2022

need a minus sign which corresponds to the ferromagdl
netic order (keep in mind that dy> 0). Now we are readyto mark the direction of the dipoles on the sketch, see thefigure below.

The work needed to pull out a magnet is found similarly to the part (C.1):
2⃗ 5 · mx̂ = 3µ0 m = 15 mJ.
W =B
4πδ 3
⃗
2
⃗ 3x = µ0 ma) B
4πl3 (3 cos α − 1)µ0⃗b) B4x = ± 2πδ3⃗ 5 = 4B
⃗ 3x + 2B
⃗ 4x
c) B

µ
m
1
⃗ 5 = 0 3 − ± 2 x̂.
d) B
2πδ
2
d ⃗
e)F5y = dy
B3 · m⃗dl 3µ0 mf) F5y = ∓ dy
16πδ 3g) F5y attractiveh) correctly marked 12 arrowsi) ferromagnetic
20mj) W = 3µ4πδ 3k) W = 15mJ2

0.2 pts
0.1 pts
0.1pts0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts

Remark: ± signs are not required as long as the correct
sign corresponding to the ferromagnetic order are used:meaning, + sign in f) and - sign in d). No marks for e) if
the score for d) is zero.

Theoretical problems: solutions. Language: English

IPhO 2022

T2: James Webb Space Telescope (12 pts)

so the diameter of the image isSome general notes for entire Theory 2 markingdiAn equation which is dimensionally correct, but missing a multiplicative factor or having a single transcription error from a previous equation, will receive a deduction of -0.1 pts.

=

(1.7 × 1011 m)(130m)=(89 ly)(3 × 108 m/s)(365 d/y)(86, 400 s/d)=

2.6 × 10−5 m = 26 µ m.

Marking scheme:

An equation which is dimensionally incorrect or one
which has more than two transcription errors will receive no points.
Follow on errors are not transcription errors; the only
penalty will be in the first occurrence of a mistake, except in the case of a dimensionally incorrect equation,
which still receives no points, even if a follow on error.
There are two follow on caveats below.
If an error in an equation trivializes the remainder ofthe problem, then no additional points after that should
be awarded. For example, if a student is computing
counts, and they arrive at the incorrect answer of zero,then they should not get future points that compute intensity, density, uncertainty, as these would all become
trivial.
If an error in an equation makes the remainder of aproblem physically unrealistic, then they should get nopoints for any requested numerical results, but they cancontinue to get points for theoretical equations. For example, if a student has an extra factor of 100, they can
get points for derivations, but if asked to find a temperature they will not get points for reporting 100 times theactual temperature. They will also not get points for reporting the correct actual temperature, because it will
not be consistent with their theory.
If an equation can be implied to have been used, thenthe assumption is that it did exist and would get points.
For example, writing Eq. 7 without explicitly writing
Eq. 6 would get points for both equations, subject to error rules above.
In places on the mark scheme there are a range of acceptable answers, and in places the range is divided into
two possible ranges, a close range for full points, and alarger range for partial points. This might appear like
this:
35µm ≤ dd ≤ 47µm
20µm ≤ dd ≤ 90µm

0.2 pts
0.1/0.2 pts

which means that they get 0.2 pts if they are within the
narrow range, but only 0.1 pts if they are outside the narrow range but still within the larger range. They would
never get 0.3 pts in this scheme, so don’t double count!

0.2 pts
0.2 pts
0.4ptsUnits must be shown for a numerical result to get
points; writing the correct answer without showing
work also receives full marks for this problem.
2. Diameter of central maximum
The angular radius of the central maximum isθmin = 1.22

λ
D

(4)

λ = 800nm is given in the problem
D is the aperture size, which is the primary mirror,
or π4 D2 = 25 m2 , so
D = 5.6m
The diameter of the central maximum is thendd = 2θmin f = 2.44

λf
λf = 1.22 p
D
A/π

(5)

The numerical value isdd(8 × 10−7 m)(130 m) =(5.6 m)=2(1.22)=

4.5 × 10−5 m = 45 µ m.

dd = 37µm is also acceptable (omitting the factor of
1.22 is okay).
Marking scheme:
correct formula Eq 5
Aperture D = (5.6 ± 0.2)m
35µm ≤ dd ≤ 47µm
sum

0.1 pts
0.1 pts
0.2 pts
0.4ptsNo penalty for ignoring factor of 1.22, so check theirmath. Units must be shown for a numerical result to
get points; writing the correct answer without showing work also receives full marks for this problem.
3. Equilibrium temperature of the detector at the location of the image?

Part A: Imaging a Star (1.8 pt)

The radiant power from the star is

1. Diameter of image
The ratio of diameter do for an object at a distanceDo ≫ f and an image diameter di is given by
di
f
=
,
do
Do

correct formula Eq 3
di = (26 ± 1) µ m
sum

(3)

(6)

Pg = 4πro 2 σTg 4
The intensity at the location of the scope is
Pg
=
Ig =
4πDo 2



ro
Do

2
σTg 4

(7)

Theoretical problems: solutions. Language: English

IPhO 2022

This is collected onto the mirror with area A and focused on a single spot of radius ri , so that the powerincident is

2
 2
ro
ri
Pi = AσTg 4 = A
σTg 4
(8)
Do
f

Part B: Counting Photons (1.8 pt)
1. Temperature of source
We are interested in the slope of the graph, which isslope =

(3) − (−1)= −100 K(0.111/K) − (0.151/K)

But at the image we have an equilibrium temperature of
Pi = aσTp 4 ,Since this is a characteristic temperature, it is at least
a partial answer to the problem.

where a = πri2 , so

The value ofaσTp 4 =

ri
f

2

|

AσTg 4

So the value ofor, ignoring diffraction,

Tp =

A
πf 2

 14

Tg ≈ 530 K



di + dddi2
a ≈ 7.46a

This means the actual pixel temperature will be
Tp,correct =A(7.46)πf 2

 14

Tg ≈ 320K.

(11)

Marking scheme:
power of source, Eq 6
intensity at mirror, Eq 7
power of image, Eq 8
correct for diffraction Eq. 10
Either Eq. 9 or Eq. 11
numerical resultsumMarking scheme:
slope of graph = −100K
Tgraph = 230K
Tsource = 1380K
sum

(10)

where the actual ratio depends on answers above.
∆Eg= 6 × 230K = 1380K
kB(9)

When considering diffraction the actual area of thestars’ image is larger,
a′ =

∆Eg
| = ln 10 × 100 K = 230K
6kB

0.2 pts
0.2 pts
0.2 pts
0.1 pts
0.1 pts
0.2 pts
1.0 ptUnits must be shown for a numerical result to get
points; the answer T ≈ (320 ± 10)K for including
diffraction or T ≈ (530 ± 10)K for ignoring diffraction must be consistent with their approach. Checkthe number, since the ratio in Eq 10 depends on theiranswer to A.2

0.2 pts
0.1 pts
0.1 pts
0.4 ptWriting either temperature correctly implies theyfound the slope of graph, and would get the +0.2 pts.
Just writing Tsource = 1380K gets full marks, as it really is possible to solve this in one’s head.
Order of magnitude T = 103 K will get full marks, andno work needs to be shown.
As this is order of magnitude, the following final answers will get full marks: T = 600K, T = 1000K,
T = (1380 ± 10)K, T = 1500K. Other numbers in
the range 500 ≤ T ≤ 1500 that are more precise
than these answers ought have a problem score of
no more that 0.2 pts, and must show working that
supports their answers!
2. Write an expression for the total count uncertaintyσt
The three uncertainties are
σr
and
σd =

Students must present a symbolic equation in theirsolution.

andWriting Eq. 11 without showing any other work receives 0.8 pts; Writing Eq. 9 without showing anyother work receives 0.7 pts.

and then

σp =

√

id τ

√

pτ

σt 2 = σr 2 + (id + p) τMarking scheme:
correct error for dark current
correct read photon
added in quadraturesum

0.1 pts
0.1 pts
0.2 pts
0.4 ptTheoretical problems: solutions. Language: English

IPhO 2022
Writingσt = σr +√id τ +√pτ

only gets +0.1, instead of the quadrature +0.2
Forgetting the read error term is a -0.1 pt deduction.
Correct dark current and photon count errors in final answer are acceptable evidence for those points;
it is not necessary for the student to explicitly state
what is what.
3. Determine the photon count for a signal to noise ratio of S/N = 10.
At a temperature of T = 7.5K, the dark current is id =5 electrons/second. This gives a total dark current
count ofid τ = 5 × 104
Answers in the range id = 5 ± 1 will be accepted forfull marks.
Let P be the photon count. Then
P = 10σtso
P 2 = 100 σr2 + id τ + P



(12)

with solution P ≈ 2290, and a rate of p = 0.229 photons per second.
Marking scheme:
id = (5 ± 1) e/s
1 ≤ id ≤ 10Eq 120.206 ≤ p ≤ 0.25
0.10 ≤ p ≤ 0.33
sum

0.2 pts
0.1/0.2 pts
0.1 pts
0.2 pts
0.1/0.2 pts
0.5 ptThey only get the points for p, the count rate, if itagrees with their assumption for id , so check themath!
Writing only the absolute counts P instead of therate p would get 0.1 pts for 2060 < P < 2500 and nopoints if outside this range.
Ignoring σr does not incur a penalty, as it is relatively
small.
4. What is intensity of source?
The near-infrared photons have an energy of Eg =2.3 × 6kB T , soEλ = (1380 K)(1.38 × 10−23 J/K) = 1.9 × 10−20 JThis is not an order of magnitude question like B.1
The energy received every second is
E = (0.23)(1.9 × 10−20 J) = 4.4 × 10−20 J
and the incident intensity on the primary mirror is
then
I=

E/t
(4.4 × 10−20 J/s)=
= 1.8 × 10−22 W/m2
A
(25 m2 )Marking scheme:

Eλ = (2 ± 0.1) × 10−20 J
Forgetting ln 10 factor
I = (1.8 ± 0.2) × 10−22 W
sum

0.3 pts-0.1 pts
0.2 pts
0.5 ptIf they forget factor ln 10, then the correct intensity
would be (7.8 ± 0.2) × 10−23 W. They only get the ln 10
penalty once!

Theoretical problems: solutions. Language: English

IPhO 2022

Part C: The Passive Cooling
1. Find expressions for the temperatures of first andfifth sheet
This is a cleaned up version of an “ideal” solution
Let Qi represent heat flow off of a surface, and Qij
represent the heat flow difference off of two surfaces that are facing each other.

A student will need to consider energy balanceacross any one sheet:′
qi−1,i = qi,i+1 + qi,i+1(23)

basically stating that the net flow into sheet i fromsheet i − 1 must equal the net flow out of sheet i toeither sheet i + 1 or into space.
Substitute in Qij ,αQi−1,i = αQi,i+1 + βQi,i+1

The student needs to consider the three types of differences below:

orBetween sun and first sheet:I0
Q01 = ϵAσ
− T14
σ

The relation for sheet 1 is a little different:(13)Qi−1,i =which is not the net heat flow between the sheets, it
is merely a convenient expression to use later.

(25)

and so is the relation for sheet 5:
q4,5 = Q5,6

Between any two adjacent sheets:
(14)

(24)

Qi,i+1

q0,1 = Q0,1 = αQ1,2 + βQ1,2which is the net heat flow into sheet 1 from the sunside.
Qij = ϵAσ Ti4 − Tj4 ,

α+β
α

(26)

What will eventually matter most is
Q5,6 =α4Q0,1(α + β)4(27)

Between last sheet and the cold, cruel vacuum of
space:

Q56 = ϵAσ T54 ,(15)

Now use the recursion of Eq. 24 to sum up the six Qij
terms in Eq. 22:

which is the net heat flow out of the far side of thelast sheet.

with the constant k defined as

From the problem text, the flux emitted by one sheet
and absorbed by an adjacent sheet is

k = 1+

qi = αQi
so that the net heat flow flux out of one sheet absorbed by the adjacent sheet isqij = αQijand the flux ejected into space between two sheetsis′qij= βQijThis doesn’t affect the marking, but the approximation being
made here is that β is the same for all four pairs of adjacent sheets.
This makes the math solvable, and was the explicit assumptionthat the students were told to make.

A student will need to recognize that


I0
4
− T1
Q01 = ϵAσσ

Q12 = ϵAσ T14 − T24
Q23 = ϵAσ T24 − T34
Q34 = ϵAσ T34 − T44
Q45 = ϵAσ T44 − T54
Q56 = ϵAσ T54

(28)

1
α
α2
α3
α4
+
+
+
+
2
3
4
α + β (α + β) (α + β) (α + β) (α + β)4(29)

Substitute the expression for Q0,1 back into Eq. 13
and get
s 
 r
1
4 I0
4 I0
T1 =
(k − 1)
(30)
1−
=
σ
k
kσ
and the into Eq. 27 and Eq. 15 to getr
α 4 I0T5 =α + β kσ(31)

which can also be written elegantly as
r
α 4 1
T5 =
T1 .
α+β k−1
Marking scheme:

(16)
(17)
(18)
(19)
(20)
(21)

can be summed to giveQ01 + Q12 + Q23 + Q34 + Q45 + Q56 = ϵAI0

kQ0,1 = ϵAI0 ,(22)

Net flow into sheet 1 Eq 13“Net” flow sheet i → j Eq 14
Net flow out of sheet 5 Eq 15
Sum to eliminate sheet temps Eq 22
Generic Energy flow Eq 23
Recursion for Qij Eq 24
Sheet 1 Energy flow Eq 25
Sheet 5 Energy flow Eq 26
Simplify sum Eq 28
Find k Eq 29
Final Expression for T1 , Eq 30
Final Expression for T5 , Eq 31
sum

0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
2.4 ptIPhO 2022

Theoretical problems: solutions. Language: English• In most cases a single mistake in an equation thatis still dimensional correct will get 0.1 pts for theequation. Making the same mistake multiple times
is not a follow on error, and would be penalized
every time.
• Any equivalent to Eq 14 would get the 0.2 pts.
• Any attempt to balance energy flow on a genericsheet like Eq 23 that is dimensionally correct andreasonable given their presentation would get the
0.2 pts• Since sheet 1 and sheet 5 have a different energy balance approach, they must show those separately to get those points.
• It is possible to arrive at Eq 28 based on dimensional analysis alone. A student who writes someform of Eq 28 without clear justification would get
points for Eq 22 and Eq 28. They could get fullmarks for final sheet temperatures if it is consistent; if they did, then they would probably also getat least partial points for Eq 13 and/or Eq 15. They
would need to introduce one more unknown constant to have defined Q56 = k ′ Q01 . The maximum
points I would expect with this approach is 1.2 pts.
• k in Eq 29 is allowed a single error for 0.1 pts. Twoerrors is no points.
• Failing to include the back flux of Eq 14 is only apenalty on that equation but would be zero points,
as it is a serious error. That means writing theequivalent of Qij = ϵATi4 is zero points! The work
after this would have a follow on error that would
need to be traced.
Corrections to some of the above formula
Andres Poldaru, leader from Estonia, pointed out
an inconsistency in part of the Eq. 23 above derivation. The above equation does reflect the simplified
schematic diagram of energy flow and loss on thequestion paper, but does not properly reflect the energy balance on an individual sheet. It should read,for sheets 2, 3, and 4, as
qi−1,i = qi,i+1 + 2BQi
where B is the physical fraction of energy lost to
space from one side of a sheet, and Qi = ϵAσTi4 . A
similar correction would exist for sheet 1 and 5, except that the factor of 2 in front of B would be 1, as
Q01 = q12 + BQ1 and q45 = Q56 + BQ5
The factor of 2 isn’t the problem, it is instead that it is
now difficult to align the expression for energy loss
to space between sheets with the energy lost to spaceof a specific sheet. As such, this approach creates amuch messier solution.
There is no convenient way to solve these corrected
five equations without iterating, which is the motivation for some of our substitutions and approximations above, or by creating a five by five matrix anddiagonalizing. At least some students attempted to
do it this way.
It does not change Eq. 28, it does change the expression for the convenient constant in Eq. 29. It alsochanges the fifth sheet heat flow, Eq. 27. Because thevariations of what students can cook up while trying
to reconcile difficult physics can be numerous, all of
the possible results are not presented here, as they
depend on what assumptions the students opted tomake.
It does not affect the approach to C.2 or the scoringof numerical bounds on C.3
Additional Marking Guidance:• A generic energy flow expression like Eq. 23that is consistent with what the student has presented will get 0.2 pts. The energy flow expression can have one reasonable simplifying assumption/approximation, so long as it does not make ittrivial, for no penalty.
• A recursion relation like Eq. 24 that is consistent
with what the student has presented will get 0.2
pts. The recursion relation can have one reasonable simplifying assumption/approximation, solong as it does not make it trivial, for no penalty.
This likely means that the student introduced a
few constants to keep it clean. That’s okay. Errors in the recursion relation that are unsupported by statements of approximation or reasonable physics, or are just plain bad math, would
have a penalty of -0.1 pts.
• Equations 13 and 15 must be consistent with thestudent’s statement of Eq. 23, or they would get apenalty of -0.1 pts. In short, pick specific approach.
• Eq. 29 must be consistent with the student’s approach; similarly, the final expressions for T1 andT5• Attempting to write this problem as a matrix but
not being able to solve it is equivalent to writing arecursion relation for 0.2 pts and simplifying thesum for 0.2 pts; in theory they have already gotten
much or most of the points above that. Not solving
the matrix, however, will have a maximum scoreof 2.2 pts, assuming everything else is there.
• As a reminder, don’t mix and match grading
schemes; follow an approach that is self consistent, and if more than one scoring approach isvalid, select the one that gives the higher score.
Original Solution
Don’t use this, eh?
Start with a statement of net energy flow q01 into the first sheet
from the sun:()
q01 = ϵA I0 − σT14
(32)
where A is the area of the sheet, ϵ is the emissivity, σ is the StefanBoltzman constant, and T1 is the temperature of the first sheet.
Now consider the space between two sheets i and j. Each sheet
radiates an energy flow
ϵAσT 4
toward the other sheet, but a fraction β is ejected into space out
the gap.
We have defined α as the fraction emitted from one sheet that isabsorbed by the other sheet, so the net energy flow from sheet i
into sheet j is()
qij = αϵAσ Ti 4 − Tj 4
(33)
There is also a lost fraction emitted into space from between the
sheets, given by
(
)
β
′
qij
= βϵAσ Ti 4 − Tj 4 = qijα(34)

Theoretical problems: solutions. Language: English

IPhO 2022

Don’t make the mistake of assuming that α+β = 1, as some of theenergy emitted from a sheet could be reabsorbed by that sheet.
Finally, write an expression for the net thermal radiant energy
flow into space, with an ambient temperature of Tspace = 0, fromthe far side of the fifth sheet.
()
q5s = ϵA σT54 − σTs4 = AϵσT54(35)
Write each of the Eq. 33, above in the form
1
qij = Aϵσ(Ti4 − Tj4 ),α(36)

and then sum up the terms from Eq. 32, the four from Eqs. 36, andEq. 35:q01 +1(q12 + q23 + q34 + q45 ) + q5s = ϵAI0
α

(37)

so

)4
ϵAI0
α
α+β
k
or
√
α 4 I0T5 =α+βkσ
which can also be written elegantly as
√
α 4
1
T5 =
T1 .
α+β
k−1
(

AϵσT54 =(43)

As this part of the question is complex, with multiple ways to go
wrong, and many opportunities for approximations, the marking
scheme will be necessarily convoluted.
Some expected mistakes:(a) Failing to account for the back flux of energy. This wouldbe
I0 = 2σT14

as all of the Ti terms cancel out on the right!
Now consider a schematic of the energy flow below

and then

ασT14 = 2σT24 ,and so on, concluding with
I0 = σ

( )4
2
T5
α

or
T5 =

From energy conservation, the net flow into sheet one from thesun and the net flow out of sheet one toward sheet two or ejectedfrom gap is′
q01 = q12 + q12,(38)′ is the part emitted into space from the gap.
where q12

Combine with Eq. 34 and
(
)
β
α+β
q01 = 1 +
q12 =
q12
α
α

(39)

Similarly, for the remaining pairs of sheets,
(
)2
α
α
q12 =
q01 ,
q23 =
α+β
α+β
and
q34 =

α
q23 =
α+β

and
α
q34 =
q45 =
α+β

(

(

α
α+β
α
α+β

(41)

Combining Eq. 32 with Eq. 41,()
ϵAI0= ϵA I0 − σT14
k

T1 =

(

1
1−
k

From above,

(
q5s =

√

)
=

α
α+β

4

Area of gap:q01 ,is a convenient constant.

I0σAssuming students grab the hint about effective absorptive areas, then expect

α=

)4

1
α
α2
α3
α4
+
+
+
+
2
3
4
α+β
(α + β)(α + β)(α + β)(α + β)44

2. Find α and β

q01 ,

where

√

(c) Incorrectly resolving β and α.

I0(k − 1)
kσ

)4
q01 .

p
Asheet(44)

Assume that the probability of being absorbed by asheet is the ratio of effective areas

The sum on the left side of Eq. 37 can then be written asso

ϵI0 = σT14

Area of one sheet A

Finally, for the fifth (last) sheet all of the net energy flow in fromthe fourth sheet must be completely ejected into space on the dark
side.
(
)4
α
q5s = q45 =
q01 .
(40)
α+β

k=1 +

(b) Inconsistent treatment of emissivity
The most likely error is of the form

Agap = 4h)3kq01 = ϵAI0

α
T1
2

(42)

ϵAsheet2ϵAsheet + Agap

(45)

This result yields α = 0.3.
Assume the probability of ejection is a ratio of effective areas
Agap
β=
(46)
2ϵAsheet + Agap
This result yields β = 0.4.
Marking Scheme:
Gap area Eq 44
Estimating α Eq 45
Estimating β Eq 46
Factor of 2 for A in both
Weighting A by emissivity in bothFinding α0.25 ≤ α ≤ 0.35
Finding β
0.3 ≤ β ≤ 0.83
subtotal
Find a better β
sum

0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
1.4 pt0.2 pts
1.6 ptTheoretical problems: solutions. Language: English

IPhO 2022• “in both” means that to get the points they musthave used the factor of two and the emissivity bothtimes; if it is missing from one, they get 0.1 pts forthe equation it is present in.
• Find α and β means that it is consistent with own
work.
• Assuming 2α + β ≈ 1 with proof would mean they
only need to find either α or β, and they would get
all of the points upon finding the other one. The
highest possible subtotal score in the case would
be 1.4 pts. Proof can be simple, however, like saying “two sheets, equal probability of being transmitted or absorbed into the other.”• Assuming 2α + β ≈ 1 without proof would mean
they only need to find either α or β, and theywould get 2/3 of the points upon finding the otherone. The highest possible subtotal score in the casewould be 1.2 pts.
• Assuming α + β ≈ 1 stating a “reasonable” proof
would mean they only need to find either α or β,
and they would get 2/3 of the points upon finding
the other one. The highest possible subtotal scorein the case would be 1.2 pts. Stating “energy conservation” is reasonable, though incomplete.
• Assuming α + β ≈ 1 without any proof would
mean they only need to find either α or β, and theywould get 1/2 of the points upon finding the otherone. The highest possible subtotal score in the casewould be 1.1 pts.
• Finding α and β means that it is consistent withown work.
• Read the special note about finding a better β below to understand the last 0.2 pts.
Special note

Any who correctly does this gets those 0.2 pts. If they
make a single mistake, but still end up with
1 − α > β > 1 − 2αthey can still get 0.2 pts. If they make two or three
mistakes, but still end up with
1 − α > β > 1 − 2αthey can still get 0.1 pts. They only get these points
for an effort to deal with our odd definition, and recognizing that the back flux is positive for ejection
from the gap. Just writing a different β without justification doesn’t get these “special” points.
In the event that a student derives
β=

5 10
5 ′
β = − α333Then their minimum score for C.3 should be 0.8
pts, then subtract off 0.1 pt for every error in theirderivation if their answer is close. After that, lookback at their work on estimating α or β alone, andadd on half points for any success, up to 1.4 pts (or1.6 pts, if no mistakes). The score they get for C.3
would be the larger of the two scoring approaches.
Original Solution
These might still apply in some cases; the first path
was rewritten above, so not included, and the second path assumed reflective sheets at angles, so wasdeleted. Choice C is a variation that can yield a correct value for β, but it would need to be combinedwith some other approach to find α.
Choice C: Estimate the radiant flux from the gap

Our assumption is that the rejected heat can be written as
qij = βϵσA Ti4 − Tj4

Assuming that the enclosed volume is a black body in equilibrium, which it isn’t, at a temperature equal to a quartic averagingof the two temperatures: 12 (Ti4 + Tj4 ). Then the energy is radiatedout of the area according toThis is certainly true, but β would be infinite in thecase of Ti = Tj . It would have been better to write
qij = β ′ ϵσA Ti4 + Tj4

1
qlost = σAg (Ti4 + Tj4 )2which would follow the energy conservation rule
2α + β ′ = 1 if ϵ ≪ 1.
In fact, β is really what the student is finding in theapproach above.

where Ag is the area of the gap, given by√
Ag = 4h A
But energy was entering the region at the rate
qin = ϵσA(Ti4 + Tj4 ),′Assuming that the temperatures of adjacent sheets
are related by
Tj = γTithen
β=β′1 + γ

41 − γ4In our case,
γ4 ≈(100K)(400K)

which means55 10
β = β′ = − α333is the best estimate; in our case, we expect β = 0.67.

so the fraction lost is
β=

2h
Ag
= √ = 0.7
2A
ϵ A

Marking Scheme:
Estimating flux out of gap
Exact flux into volume
Correct estimate of gap area
Finding β0.65 ≤ β ≤ 0.75
sum

0.2 pts
0.2 pts
0.2 pts
0.1 pts
0.1 pts
0.8 ptThe bounds on allowed values for β are smaller in this approach,because there really is only one reasonable answer.
Look back at the full solution to see how to score estimates for α
based on this β.
Note that this approach has fewer possible points, as the expression for the flux out of gap makes an assumption that is based onunchecked physics.
Choice D: Another Approach?

Theoretical problems: solutions. Language: English

IPhO 2022

Surely there will be some creative students who show other approaches. We will try and expand the marking scheme to recognise these approaches as soon as they occur. A rough guide for
an incomplete approach is
Tentative Marking Scheme:
Relevant correct physics equation, each
Reasonable approximation, each

0.2 pts
0.1 pts

The maximum possible is still 1.6 pts.
An equation is only relevant if it can be argued that it would leadto an answer to the question within the bounds of the approachthat they are following. For example, don’t award points for bothcounting bounces and effective surfaces, unless each equation
contributes to a unified approach that would lead to the answer.
Find the most rewarding approach, and award points for that lineof reasoning.
If a student only finds one of α or β, then they get 0.2 pts for the
first. The marking scheme assumed they would look for α first,
but they might have looked for β, and only found that.
Be very careful with mixing and matching approaches!
A student will not get half the points for one approach plus half
the points for another approach if they attempt, but don’t succeed, with both approaches. They will be awarded the higher of
the two scores, not the sum.

3. Numerically determine the temperature of sheet 1
and the temperature of sheet 5.
The solar intensity is I0 = 1360 W/m2 , the background temperature of space is Tb = 20 K and is negligible.
Assuming a student does C.1 correctly, and uses 2α +β = 1, then
β
0.3
0.4
0.5
0.6
0.7

α
0.35
0.3
0.25
0.2
0.15

T1 (K)
383
380
376
373
369

T5 (K)
120
102
83
65
48

The other bound is α + β = 1, in that case:
β
0.3
0.4
0.5
0.6
0.7

α
0.7
0.6
0.5
0.4
0.3

T1 (K)
370
368
365
363
361

T5 (K)
189
165
140
114
87

The numbers agree well with the theoretical performance of 320 K and 90 K. Some of the major differences are explained by different coatings on different surfaces, a temperature and wavelength dependence on emissivity that is designed to reflect visible
light from the sun while radiating infrared on thesunside of sheet 1, and the sheets are not uniform
temperature.
Marking Scheme:
T1 consistent with own formula
250K ≤ T1 ≤ 400K
T5 consistent with own formula
45K ≤ T5 ≤ 200K
sum

0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.4 pt

The grade depends on self consistency with the previous work, so the numbers must be checked!
Note that here is a case where follow on errors could
be penalized twice; students should recognize thatan answer is not reasonable, as T1 should be on theorder of the temperature of the Earth, and that T5
ought to have shown significant, but not incredible,
cooling.

Theoretical problems: solutions. Language: English

IPhO 2022

Part D: The Cryo-Cooler

Gas Parcel

1. What state variables change?
(a) In order to force the gas through the plug,
which offers up considerable viscous friction,
P1 > P2 ; it is this pressure difference that is thesource of the force.
(b) Viscous friction is dissipative, and the energyof moving through the plug is not allowed to
go anywhere else, so the internal energy of thegas must increase as it moves through the plug,and then U1 < U2 .
(c) Though no heat is gained or lost, this is not a
constant entropy process; that can be seen because it is an irreversible process. As such, S1 <S2(d) Since the process of moving across a pressure
gradient imparts kinetic energy to an object, it
is expected that the fluid velocity on the right
will be higher than the left; in fact, we are told
this is marginally true. Since mass is conserved,
the volume of a mole of gas on the right mustalso be higher than the volume of a mole on theleft, and V1 < V2 .
(e) The correct answer is T1 ?T2 . If this were an
ideal gas, one might expect T1 ≤ T2 since U ∝ T .
But this is not an ideal gas, and U will be a function of temperature and density. As such, it
is not possible to know the comparative relation between T1 and T2 . That’s the whole pointof this problem, and the challenge of trying to
make liquid helium.
Marking scheme:

Porous Plug
The work that the surrounding gas in region 1 doeson the gas pushing it into the plug is
W 1 = P1 V 1
because the pressure is constant, and the effective
change of volume is V1 . Similarly, when the gas enters region 2 it must displace a volume V2 of gas that
was already there, so
W2 = −P2 V2The net work is then(47)

Wnet = P1 V1 − P2 V2Since there is no heat exchanged,
U2 − U1 = ∆U = Q + Wnet = P1 V1 − P2 V2(48)

which implies∆U = U2 − U1 = P1 V1 − P2 V2 .
Upon rearranging
U 2 + P2 V 2 = U 1 + P 1 V 1and therefore
U + PVFor each correct responsesum+0.2 pts
1.0 ptExplanations by the students are not needed.
2. A mole of gas at P1 , V1 , T1 , U1 enters the porous plugfrom the left, and that mole of gas exits the porous
plug on the other side at P2 , V2 , T2 , U2 .
Consider first a control volume approach
The figure below shows the motion of a mole of gas
through the plug; the mole is shown in pink. Gas tothe left of the mole pushes the mole through the plugwith a constant force P1 A through a volume V1 .

is a conserved quantity.
Marking scheme:
Compute correct W1
Compute correct W2
Write energy law, Eq 48
Show U + P V conservedsum

0.1 pts
0.1 pts
0.2 pts
0.2 pts
0.6 ptConsider instead a differential approach
Another way to look at this problem is to focus ona differential sample of gas as it moves through theplug.
The figure below illustrates thisGas Parcel

Porous Plug
The mole of gas moves through the plug to the righthand side, in the process pushing on the air to theright of the mole with a constant force P2 A, througha volume V2 .

The total energy of parcel of molar size δm has two
relevant energy terms: the internal energy δU andthe bulk kinetic energy δK. It has a volume δV .
These four quantities are extrinsic, but to simplify
notation, we will drop the δ. It’s still there, just invisible.
For simplicity’s sake, assume a cylindrical shape tothe parcel, with an end cap area δA and a length
dx. Once again, we will drop the δ. There are three
forces that act on the shape, one associated withpressure on the left end, one associated with pressure on the right end, and frictional force associatedwith viscosity against the walls of the container.

Theoretical problems: solutions. Language: English

IPhO 2022

Since this is a parcel of differential length dx, thenet force associated with the pressure difference between the ends is
Fends = −VdPdx

where V is again the volume of the cylinder.
But this force is (mostly) balanced by the viscous
frictional force Fwalls with the walls of the sponge;
these two forces effectively add to zero. In fact, it isthe viscous forces with the wall that cause the pressure gradient across the sponge.
The bulk kinetic energy of the parcel does notchange significantly as it moves through the sponge.
This is seen in that the bulk speed of the gas doesn’tchange significantly as it moves through the sponge.
The problem with this approach is that the system
is not in thermodynamic equilibrium; the process is
not reversible, so it is not possible to attach well defined state variables. This means that(49)

dU = T dS − P dV

is not a function that can be integrated; in fact, dS ̸=
0 from the previous part of the problem. Arguingthat V dP = −T dS is rather handwavy, and resolving
this actually requires considering a control volume
approach.
Still, the energy conservation ideas still hold true,
even if thermodynamically poorly defined, so
dU = −P dV − V dP
since the part associated with −V dP doesn’t change
the bulk kinetic energy, and instead dissipates into
internal energy of the gas.
The result is thatdU = T dS − P dV
and then requiring constant entropy so that dS = 0,
and then


∂U
P =−
(50)
∂V S
which are the negative slopes of the constant entropy curves on a U − V graph.
Another approach to find pressure is to consider aline of constant U , then


∂S
P
=
T
∂V U
Then


U + PV = U −

∂U
∂V


V
S

is the conserved quantity.
Now −(∂U /∂V )S is measured only at the point V1 , U1 ,and is the slope of the tangent line to the constantentropy curve. Following that tangent line back a
distance V takes it to an intercept with the U axis,
and that intercept is then the conserved quantity.
More mathematically, define a function H
H = U + PVthen

U = H 2 − P2 V

is the equation of a line,U = H2 +

∂U
∂V


V

(51)

S 2

with the U intercept equal to the conserved H2 .

dU = −d(P V )An estimate can be made visually, but it is difficult tobe accurate. Try constructing a line from the point
V2 = 0.120, T2 = 7.5 that is tangent to the local isentrope, and the result will intercept the U axis. This
result is somewhere around 40. This is shown ingreen below.

or
U + PVis a constantMarking scheme:
Traditional δW = −P dV
Bulk kinetic δK = −V dP
Explain where δK goes
Differential Eq 49
integrate U + P V constantsum

3. One can find pressure on this graph by applying

0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.1 pts
0.5/0.6 pt

Because of the many subtle traps, this approach will
not get the same number of points as the control volume approach.
Writing dU = −P dV and integrating to find U + P V
is constant gets only 0.2 pts. This is because there
are several errors: the differential is poorly definedwithin the sponge; because the state variables arepoorly defined; P is not a constant; so you can’t actually integrate it; and the work done in this case is
not correctly computed. Four wrongs don’t make aright.

Now to improve the result.
Draw a line out from 39 that is tangent to the nearest isentrope to V2 = 0.100, T2 = 7.5; draw anotherIPhO 2022

Theoretical problems: solutions. Language: English

line out from 41 that is also tangent to the nearest
isentrope to V2 = 0.100, T2 = 7.5. These are shown inpurple below.
Measuring the distance with a ruler, find the fractional distance between the two purple lines to thepoint V2 = 0.100, T2 = 7.5 along the highlighted green
line. It is about 75% the way from the bottom purple
line. This means that the conserved quantity oughtbe 75% the way up on the highlighted blue section
on the graph. A line connecting the two is shown ingreen.
This point is about 41 kJ/kg. The actual value for theconserved quantity is U + P V = 40.7 kJ/kg.
Students don’t need to draw every line, as with astraight edge one can find the tangent that maximizes the temperature T1 by shifting it around visually.
Line starts from student’s H
Line intercepts an isentrope
The isentrope matches max T1
Stated T1 within 0.5K of student’s construction
10K ≤ T1 ≤ 12K
sum

5. Using the slope of the line from the conserved quantity to the maximum temperature point, computethe pressure.

Marking scheme:
Pressure formula stated, Eq 50
Tangent intercept concept
A first estimate for H
Upper bound for estimate set
Lower bound for estimate set
Interpolated estimate set
40.5 < H < 41.0
40.2 < H < 41.2
sum

0.2 pts
0.2 pts
0.2 pts
0.1 pts
0.1 pts
0.8 ptUsing the results from above,
0.2 pts
0.4 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2/0.2 pts
0.1/0.2 pts
1.4 ptAs the task asks for a graphical construction, and it
is not possible to construct an accurate tangent to
the isentrope at T2 = 7.5K based on a single line, students must do something to improve or verify the
result, even if it is correct on the first guess. Hencethe upper and lower bound approach and interpolation, or something equivalent.
4. Draw a series of radial lines out from the conserved
point that are tangent to lines of constant entropy.
Mark the tangent point. Connect with a smooth
curve; this curve is the set of points U1 as a functionof V1 that has the conserved quantity. Look for the
maximum temperature intercept.
This happens at about T1 = 11K. If T1 is higher thanthis, it would not be possible to cool down to T2 =7.5K.

P1 = −(41) − (10)= 1.8 MPa(0) − (0.0170)

If they didn’t know to use slope by this point, they
can’t generate an answer. As such, they would already have received points for the pressure formula, and we only consider the numerical result
P agrees with the slope of the graph
1.6 MPa ≤ P1 ≤ 2.4 MPasum

0.1 pts
0.1 pts
0.2 ptTheoretical problems: solutions. Language: English

IPhO 2022

T3: Scaling laws (8 pts)

There are two processes binding the sand grains together. The first one is the force due to the surface tenNote: A correct numerical answer provided with at least sion, F1 = γl, where γ denotes the surface tension coeftwo significant figures receives full marks. Inappropri- ficient, and l — the perimeter of the “neck”; with l ∼ rg ,ate use of equality will lead to a penalty of 0.1 pts for each where rg denotes the length scale of a single grain, we obpart of the question.
tain Fs ∼ γr. The second one is the pressure force causedby the negative capillary pressure in the neck, Fp = ∆pA,
where A is the cross-sectional area of the “neck”, andTask A: Spaghetti (2 pts)
∆p ∼ γ/r. With A ∼ r2 we obtain F2 ∼ γr. Thus, the bothcomponents are of the same order of magnitude and usThis is section 2.2.2 (Statics) of the syllabus.
ing either of them will lead to the correct scaling law.
These forces press the grains against each other, hence
Consider only the left half of the spaghetti straw.
Torque balance at its right endpoint implies that the the normal force and friction force between the grains istorque applied to its right endpoint must balance out the also on the order of Fs and Fp .
Solution 1:
torque due to gravity: τ ∝ ml ∝ d2 l2 . This torque arises
from the gradient in the horizontal stress. If the typical
Based on what has been said above, the typical force
horizontal stress is σ, then the typical force is F ∝ σd2 , needed to delocate a grain of sand is Fg ∝ rg . The force
so the torque is τ ∝ F d ∝ σd3 . Hence, we obtainneeded to delocate an entire horizontal layer of sand isthen ∝ Fg Nl , where Nl ∼ A/rg2 is the number of grains in√
32 2
σd ∝ d l =⇒ l ∝ d,a layer. The force of cylinder destruction F thus satisfies
so

r
l′ =

d′
d

l=

√

F ∝ Fg Nl ∝ rg /rg2 = rg−1 ∝ Vg−1/3 ,10 · 50 cm = 158 cm.

Marking scheme:τ ∝ d2 l 2F ∝ σd2
τ ∝√
σd3
l∝ d
Answer: 158 cm

0.4 pts
0.5 pts
0.5 pts
0.4 pts
0.2 pts

Task B: Sand castle (2 pts)
This is Section 2.2.2 (statics) and 2.2.5 (hydrodynamics) of
the syllabus
Due to wetting of the surfaces of the sand grains andits large surface tension water acts like a glue for sand.
This means that all the grains need to be bound together
by air-water interface. To achieve this there needs to beneither too little nor too much water: if there is too little
water, most of the grains are dry with no surface tension binding them, and if there is too much water, almost all the grains are immersed into water, and again,
there is no surface tension binding the grains. So, theoverall strength of the buildings from wet sand depends
on the water content; we assume that for the both typesof sand, the water content is optimal, and the shape ofthe grains is statistically similar. Let us consider twoneighbouring grains connected by a water meniscus —or “neck”, as we shall be referring to it henceforth. Notethat the “neck” may extend perpendicularly to the figure plane far away; so, more specifically, what the word
“neck” will refer to is that part of the water-air interface
for which the closest two grains are the ones under consideration.

so

Ff g = (1/10)−1/3 · Fcg = 21.5 N.

Marking scheme:a) Fs ∝ rg and Fp ∝ rg
one of the two missingb) Fg ∝ rgc) F ∝ Fg Nld) F ∝ rg−1
Answer: 21.5 N

0.5 pts-0.1 pts
0.5 pts
0.5 pts
0.3 pts
0.2 pts

Notes: If the student only qualitatively explains themechanism by which the grains of sand are held together, a maximum of 0.5 pts are given. Points b)-c) are
given only if derived from a).
Solution 2:
We have seen above that grains to one side of a fictitious surface exert force per cross-sectional area on theorder of magnitude as the capillary pressure ∆p ∼ γ/rg .
In order to get the grains moving, a pressure of the sameorder of magnitude needs to be applied externally. For
both cylinders, the surface area where the force is applied is the same, hence the force scales as the capillarypressure, F ∝ 1/rg ∝ V −1/3 .
Marking scheme:
The applied pressure must be ∼ ∆p
The curvature radius of the interface is ∼ rg
Capillary pressure ∆p ∼ γ/rg
Answer: 21.5 N

0.6 pts
0.6 pts
0.6 pts
0.2 pts

Note: If the contribution of surface tension is neglected,
0.1 pts are subtracted.
Solution 3:
The compression force serves to break the surface tension bonds between sand grains.
Consider the energy E required to push a single layerof sand into the layer beneath it. E ∝ F rg , where F is theIPhO 2022

Theoretical problems: solutions. Language: English

force required and rg is the typical height of a layer (i.e.,
the typical length scale of a grain).
On the other hand, E = γ∆A, where γ is the surface
tension of water and ∆A is the total amount by which
the surface of the water in the layer stretches before all
the “water bonds” between the sand grains are broken.
Here, ∆A is proportional to the area A of a layer and isthus a constant between the two cylinders. Hence, E ∝F rg is a constant between the two cylinders, i.e., F ∝ rg−1 .
Marking scheme:d ∝ gt2
Answer: 1.5

0.2 pts
0.1 pts

Solution 1:
One way to approach the problem is to notice that constant acceleration in spaceship’s frame means a constantforce in the Earth’s frame. This follows directly from theLorentz transform for the electromagnetic field, more
specifically from the fact that when going to a frame
moving parallel to the x-axis, the x-directional electricfield Ex remains unchanged. Hence, on the one hand,the force Fx = eEx exerted on an accelerating particle ofE ∝ F rg
0.5 pts
rest mass m0 and carrying a charge e remains constantE ∝ γ∆A
0.5 ptsin the lab frame. On the other hand, the acceleration of∆A ∝ A
0.5 pts
that particle in an inertial frame moving with velocity v,
0.3 ptsF ∝ rg−1
where v denotes the particle’s velocity at a certain moAnswer: 21.5 N 0.2 pts
ment of time t, is always equal to eEx /m0 , regardless of
Note: If the student only qualitatively explains the mech- the value of t, i.e. constant in time.
anism by which the grains of sand are held together, aThose who are not familiar with the Lorenz transformmaximum of 0.5 pts are given.
for electromagnetic field can derive the above described
Solution 4:
property from the Lorenz transform for momentum andFirst of all, the force F should be proportional to the coordinates. We use again (i) the lab frame, and (ii) an incylinder’s base area A. The force required to destroy a ertial frame moving with velocity v, where v denotes thecylinder with base area A = nA0 is equal to the force spaceship’s velocity at a certain moment of time which
required to destroy n cylinders each with base area A0 . will be used as the origin, t = t′ = 0; let primes denote
As a result, F ∝ n ∝ A.
quantities in the second frame. Assuming a very short
In addition, F depends on the grain’s length scale rg time period t, we can neglect terms quadratic in time soand the water’s surface tension γ. Dimensional analysis that in the frame (ii), the momentum, coordinate and thethus gives
relativistic mass can be expressed as p′ = F ′ t′ , x′ = 0,Aγm′ = m0 , respectively; applying the Lorenz transformF ∝∝ rg−1rg
yields t = γt′ and p = γ(F ′ t′ + m0 v) = tF ′ + γm0 v. On theother hand, in the frame (i), p = γm0 v + F t; comparing
for fixed A and γ.
this with the previous result yields F = F ′ .
Marking scheme:
It appears that in either case, the spaceship’s speed willF ∝A
0.6 pts
reach almost c much faster than the travel time. Hence,
F = F (A, rg , γ) 0.6 pts
using for convenience the system of units where c = 1,
F ∝ Aγ0.6
pts
the travel distance x equals with a very good precisionrg
Answer: 21.5 N 0.2 pts
the travel time t, x = t.
What is left to do is to relate t to the proper time τ ,Note: If the student only qualitatively explains the mechanism by which the grains of sand are held together, a
dt
m0
dτ =
= dt p 2;
maximum of 0.5 pts are given.
γ
m0 + m20 g 2 t2

Task C: Interstellar travel (2 pts)
This is Section 2.5 (Relativity) of the syllabus
Let T = 50 yrs be the astronauts’ total travel time.
For maximal travel distance, the spaceship accelerates
at constant proper acceleration a = g for proper time
T /4, during which a distance of d is traveled. The spaceship then decelerates at a = −g for proper time T /4 tocome to a rest, during which another distance d is traveled. The spaceship then returns to Earth using the sameprocedure.
Notes: Formula relating acceleration to proper acceleration is not considered as a basic SR formula andtherefore if the formula is written without motivation,
0.2 pts are subtracted.
Solution 0: (incorrect)
If we ignore relativity, then d ∝ 21 gt2 ∝ g, which gives
an answer of 1.5.
Marking scheme:

upon integration we obtainτ = asinh(gt)/g ⇒ x ≈ t = sinh(gτ )/g ≈ exp(gτ )/2g.
So we conclude that the ratio of the travel distances isd2
g2=
exp(1.5gτ − gτ ) = exp(gT /8) ≈ 480.
d1
1.5g
3
Note that an exact relationship between x and t could
have been obtained by expressing the energy of thespaceship as m = m0 + m0 gx, and the momentum as
p = m0 gt. Then the Lorenz invariant (m0 + m0 gx)2 −2(m0 gt)2 = m20 yields x(x + 2/g) = t2 = sinh (gτ )/g 2 , hencex = [cosh(gτ ) − 1]/g.

Theoretical problems: solutions. Language: English

IPhO 2022
Fx is Lorentz invariantx≈tdτ = dtγγ −1 =pm0 /m
m = m20 + p2
p = m0 gt
t = sinh(gτ )/g
Answer: 480

0.4 pts
0.4 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts

Remark: if integration boundaries for distance or proper
time are wrong by a factor of 0.5, 2, 4, etc., -0.1 pts.
Solution 2:
Let w be the rapidity of the spaceship, defined as w ≡−1tanh (β), where β is the spaceship’s velocity. Then β =tanh w, the Lorentz factor γ = cosh w, and its momentum
p = m0 sinh w.
As shown by Solution 1, a spaceship experiencing a
constant proper acceleration g experiences a constant
three-force

F = m0 g =Palermo 21, 129–175 (1906)] of depicting things in x − itdiagram. The benefit of using this diagram is that therelativistic invariant x2 − t2 transforms into Euclideansquared distance x2 + θ2 with θ = it. This means thatin that diagram, we can use the knowledge of Euclidean
geometry. In particular, the Lorentz transform is now
the rotation of the Euclidean x − it-space by an anglev. Now, consider the trajectory of the space
α = arctan ic
ship; its infinitesimal arc length is icdτ , where dτ is thedifferential of the proper time, and the infintesimal rotation angle of its tangent is dα = arctan(dv/ic) = dv/ic =gdτ /ic. Therefore, the curvature radius R = icdτ /dα =−c2 /g is constant, i.e. the trajectory is a circle of radius
R. Now we can easily relate the travel distance x to thearc length icτ :



icτ
c2 
gτ
x = R(1 − cos α) = R 1 − cos
=
cosh
−1 .
R
g
c
Marking scheme:
R = const in x-ict-diag.
R = −g 2 /cmissing ‘−’
partial credit for R = icτdαx = R(1 − cos α)gτc2g cosh c − 1
Answer: 480

dp
dw
dw
g
= m0 cosh w
=⇒
=
.
dt
dt
dt
cosh w

Meanwhile, time dilation relates t to the spaceship’s
proper time τ as
dt
dw
dw dt
= γ = cosh w =⇒
=
= g.
dτ
dτ
dt dτ
Integrating yields w = gτ . Recalling that dt = γdτ , we get
the following as the total distance traveled over a quarter of the spaceship’s trip:
Z T /4d=βγ dτ =0Z T /4tanh w cosh w dτ

Solution 4: The problem can be solved by using the velocity addition formula. Let v = βc be the speed of thespaceship in the lab frame, t be the lab time, and τ —the proper time. Also, we consider a frame which moves
with constant speed v in which the spaceship accelerates
from rest:0

Z T /4=0

1sinh gτ dτ = (cosh(gT /4) − 1).
g

The answer is thus
g1 cosh(g2 T /4c) − 1
10 cosh(19.72) − 1=
g2 cosh(g1 T /4c) − 1
15 cosh(13.15) − 1
2≈ e19.72−13.15 = 480.
3
Marking scheme:ddt (m0 sinh w) = m0 gg
dw
dt = cosh w
dtdτ = cosh w
dwdτ = g

w = gτ
R T /4dβγ dτ
2 = R0T /4d=tanh w cosh w dτ
2
0
d
1
=
(cosh(gT/4) − 1)2g
Answer: 480

0.5 pts
0.1 pts
0.4 pts
0.1 pts
0.1 pts
0.3 pts
0.2 pts
0.1 pts
0.2 pts

Remark: if integration boundaries for distance or proper
time are wrong by a factor of 0.5, 2, 4, etc., -0.1 pts.
Solution 3: The problem can be also solved by using thetrick introduced in 1905 by Henri Poincaré [Poincaré,
M.H. Sur la dynamique de l’électron. Rend. Circ. Matem.

0.5 pts
0.5 pts-0.2 pts
0.2 pts
0.5 pts
0.3 pts
0.2 pts

β + dβ =Thus,

β + gdτc
1 + βgdτ
c

=β+

gdτ
(1 − β 2 ).
c

dβ
gdτ
gτ
=
⇒ β = tanh( ).
1 − β2cc

From relativistic time dilation formula we obtain
dt = pdτ
1 − β2= cosh(gτ)dτcso that the travel distance

 

Z
Z T /4dgτ
c2
gT= vdt = ccosh
−1sinh( )dτ =
2
c
g
4c
0
which leads to the same answer as before.
a) β + dβ =β+ gdτ
c

1+ βgdτcdβgdτb) 1−β 2 = cc) β = tanh( gτc )d) dt = √ dτ 21−βe) dt =Rcosh( gτc )dτf) d2 = vdt
R T /4gτg) d2 = c 0h sinh( c)dτ igTdc2h) 2 = g cosh 4c − 1i) Answer: 480

0.3 pts
0.2 pts
0.2 pts
0.3 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts
0.2 pts

Theoretical problems: solutions. Language: English

IPhO 2022

Remark: if integration in f) is done over proper time,
no points are given for f). If integration boundaries for
distance or proper time are wrong by a factor of 0.5, 2, 4,
etc., -0.1 pts.

Task D: That sinking feeling (2 pts)
(This is Sections 2.2.5 (Hydrodynamics) and 2.4.1 (Single
oscillator) of the syllabus
Solution 1: The oscillation of the half-sunk sphere isdriven by gravity. The non-damped angular frequency
depends on the gravitational acceleration and a characteristic length, which is, for a sphere, its radius r, so
ω0 ∝

p

g/r

is the only dimensionally correct possible function.
The drag force Fd depends on the sphere’s speed v[m/s], its size r [m], and viscosity of the liquid η [Pa·s].
Dimensional analysis thus gives Fd ∝ ηrv. The damping
factor is thus
Fd
ηr
β=
∝
.
2mv
m
Since the mass scales with r3 , we have1
β ∝ 2.
r
Then the relation
β2
ω2
=1− 2
2
ω0
ω0
scales as

1
β2
∝ 3
2
ω0
r

Oscillations only occur if β/ω0 < 1, so solvepr= 3 1 − (0.99)2 = 0.271
r0pa) ω0 ∝ g/r
stated without justification
effective mass ∝ r3
just the mass of the ball considered2effective
p returning force ∝ r
ω0 ∝ g/rb) Fd ∝ ηrv
no justification
Stokes without constant Kc) β ∝ 1/r22ω2d) ωβ 2 = 1 − ω
2

0.4 pts-0.2 pts
0.2 pts-0.1 pts
0.1 pts
0.1 pts
0.6 pts-0.3 pts-0.1 pts
0.3 pts
0.4 ptse) ωβ 2 ∝ r130f) Answer: 0.271

0.2 pts
0.1 pts020Solution 2: The oscillation of the half-sunk sphere isdriven by the change in buoyancy force, which is proportional to the change in displaced water volume. Thus,
the restoring force Fr ∝ r2 x, where x is the displacementof the sphere.
As discussed in Solution 1, the drag force Fd ∝ rv = rẋ.
The effective mass of the oscillation m ∝ r3 . This leads
to the equation of motion
k1 r2 ẍ + k2 ẋ + k3 rx = 0,
where k1 , k2 and k3 are constant. In the case with noviscous drag, k2 = 0, the motion is at frequency
r
k3
.
ω0 =
k1 r
With viscous drag, we can get the frequency ω by substituting trial solution x = eαt and using ω = Imα. This
leads tosk2
ω = ω02 − 22 4 ,
4k1 r




k2
k22
ω 2 = ω02 1 − 2 24 2 = ω02 1 −.
4k1 r ω0
4k1 k3 r3At r = rmin , ω = 0, so
k22
3= rmin,4k1 k3

Notes:√
1. To obtain ω0 ∝ 1/ r without dimensional analysis,note that a small displacement y changes the submerged volume of the ball by ∆V ∝ r2 y, so thechangeinpbuoyant p
force F ∝ r2 y, which gives ω0 =p
2
3
k/m ∝ r /r = 1/r.
2. To obtain Fd ∝ ηrv without dimensional analysis,note that the typical length scale l in the variationsin the velocity field of the water is proportional to r.
Thus, the viscous shear σ ∝ ηv/l ∝ ηv/r. The totaldrag force is thus Fd ∼ Aσ ∝ ηrv, where A is ball’s
area of contact with the water.
3. Alternatively, to obtain Fd ∝ ηrv, make use of the
Stokes drag relation Fd = 6πηKrv, where K is a dimensionless constant that takes into account thatthe ball is not in infinite homogeneous fluid.
Marking scheme:

and



r3
,
ω 2 = ω02 1 − minr03

giving rmin
r0 = 0.271.

Marking scheme:a) effective mass ∝ r3
just the mass of the ball considered2b) effective
p returning force ∝ rc) ω0 ∝ 1/rd) Fd ∝ rv
no justification
Stokes without constant Ke) k1 r2 ẍ + k2 ẋ + k3 rx = 0f) ω in terms of r and ω0
if not expressed in terms of ω0g) Answer: 0.271

0.2 pts-0.1 pts
0.1 pts
0.1 pts
0.6 pts-0.3 pts-0.1 pts
0.3 pts
0.6 pts-0.3 pts
0.1 pts