PC2024-P3 Best solutions and final results

Problem 3 – Scrap iron sorting

In total, 73 solutions were submitted for problem 3, of which 56 were correct. Congratulations to everyone who managed to solve it!

The Best Solution Awards were selected from the 19 correct solutions sent before the first hint was published. Selecting a few solutions from such a large number, most of which were written very nicely and typeset in LaTeX, was a difficult task. The main differences between the solutions were in how the magnetization M of a spherical ferromagnetic particle was calculated and in the textual explanations.

There were many solutions that were very nicely written and could have been prime choices for a best solutions award but eventually were not considered because of containing a minor mistake (that did not affect the final answer, see below) or gaps in derivation, e.g. not showing how the magnetization of a sphere was calculated. If you think you had a good solution, feel free to ask me by e-mail why your solution was not chosen.

A typical mistake that disqualified many otherwise good and nicely written solutions was an error by a numerical prefactor of the force acting on a piece of scrap, caused by a conceptual mistake in how the force was calculated. Namely, when calculating the force using the virtual displacement technique leading to the expression \vec{F}=\mathrm{grad} (\vec{B}\cdot\vec{m}), where \vec{m}=V\vec{M} denotes the magnetic dipole moment of the scrap particle. This is the correct formula, but only as long as you assume here the magnetic dipole moment \vec{m} is constant while taking the gradient. Indeed, it is clear that the force acting on a magnetic dipole does not depend on how the dipole would change if it were moved, and depends only on its instantaneous value. We can say that during a virtual displacement by a distance \mathrm{d} \vec{r}, the work done by the net force is \vec{m}\cdot \mathrm{d} \vec{B}, and the change of the dipole’s internal (potential) energy is \vec{B}\cdot \mathrm{d} \vec{m}, where \mathrm{d} \vec{m}=V\mathrm{d}\vec{M} is the change of the dipole moment caused by the displacement (one can imagine that the molecular dipoles are opposite charges that are connected with a spring in which case there is also the potential energy of the spring).

Note that the above given formula, \vec{F}=V\mathrm{grad} (\vec{B}\cdot\vec{M}) can be written so that there is no confusion about to which terms the derivative is applied to: in one-dimensional version, F_x=VM_x\frac{\mathrm{d} B_x}{\mathrm{d}x}, and in the generic case using the direct (also called the dyadic) product of two vectors producing a tensor as \vec{F}=V(\nabla\vec{B})\cdot \vec{M} (note that this formula has a dot product of the tensor \nabla\vec{B} with a vector, producing a vector).

And so, the Best Solution Awards go to the following contestants.

• 0.3 to Luka Passek-Kumerički – a nice solution with the sphere’s magnetization derived from the electric-magnetic equivalence based on the electric field inside a homogeneously charged sphere.
• 0.3 to Konstantin Rodionenko – a clean and compact solution with the sphere’s magnetization derived from the electric-magnetic equivalence based on the electric field inside a homogeneously charged sphere.
• 0.3 to Kutay Kulbak – a nice solution with the sphere’s magnetization guessed and shown to be correct using magnetic scalar potential (though the expression for the force is written slightly incorrectly – although the dyadic product is apparently assumed and used later correctly, the dot stands in the wrong place).
• 0.1 to Lukas Schicht – otherwise a very nicely written solution, but there is no explanation of where the expression for the magnetic field inside a homogeneously charged sphere comes from.

Final results for Problem 3

Pre-university students:

Saskia	Põldmaa	Estonia	Tallinna Reaalkool	14/1/24 15:24	2.5937
Ilan Mihelja	Ivan Hegedić	Croatia	XV. gymnasium Zagreb, V. gymnasium Zagreb	14/1/24 16:10	2.3579
Val	Karan	Croatia	XV. Gimnazija Zagreb	14/1/24 18:28	2.1436
Kutay	Kulbak	Turkey	Bahcesehir High School for Science and Technology	18/1/24 05:50	1.9763
Mihkel	Rannut	Estonia	Tallinna Reaalkool	14/1/24 18:59	1.9487
Guy	Ginzberg	Israel	Ort Irony Dalet Modiin	15/1/24 10:13	1.7716
Ralf Robert	Paabo	Estonia	Hugo Treffneri Gümnaasium	15/1/24 17:52	1.6105
Arda	Ünver	Turkey	Kocaeli Gebze Bahçeşehir High School for Science and Technology	21/1/24 17:23	1.331
Felix	Yew	Malaysia	National Junior College, Singapore	27/1/24 05:59	1.21
Matej	Gelev	North Macedonia	Yahja Kemal	27/1/24 22:11	1.1
Tornike	Tsadzikidze	Georgia	komarovi Physics-Mathematics school	29/1/24 18:38	0.9
Halil İbrahim	Yenitürk	Turkey	Ankara High School of Science	2/2/24 07:52	0.9
Oytun	Er	Turkey	Kadıköy Anatolian Highschool	29/1/24 13:00	0.9
Bahaa Aldeen	Melhem	Syria	Albasel Mixed High School	3/2/24 22:27	0.9
Darius	Graur	Moldova	Liceul Teoretic Orizont	30/1/24 09:53	0.9
Yangguang	Zheng	China	The affilicated High School of South China normal university	4/2/24 11:28	0.9
Mohamad Eid	Al-aktaa	Syria	Al saadeh school	10/2/24 18:52	0.8
Tomas	Razbadauskas	Lithuania	Kaunas University of Technology Gymnasium	11/2/24 16:46	0.7
Rares Felix	Tudose	Romania	Liceul Teoretic Internațional de Informatică București	11/2/24 19:43	0.63
Chenyun	Wang	China	Shanghai High School	17/2/24 12:58	0.63
Animesh	Pradhan	India	Modern Delhi International School	19/2/24 10:13	0.6
Jumadurdy	Nuryyev	Turkmenistan	Specialized Comprehensive Boarding School named after Saparmyrat Turkmenbashy in the city of Ashgabat	26/2/24 05:13	0.5
Vincent Karl	Willis	Austria	BRG Innsbruck APP	28/4/24 12:22	0.5
Kaustubh	Kaka	India	ALLEN	12/3/24 11:32	0.5
Andrei	Vila	Romania	International Computer High School Bucharest	7/3/24 20:49	0.5
Ruicheng	Ma	China	High School Affiliated to Harbin Teachers College	5/3/24 17:10	0.3679

University students:

Lukas	Schicht	Germany	Eberhard Karls Universität Tübingen	14/1/24 16:07	2.8665
Konstantin	Rodionenko		ITMO University	16/1/24 18:54	2.7489
Luka	Passek-Kumerički	Croatia	University of Zagreb	17/1/24 18:34	2.6305
Yi-Ting	Lin	Taiwan	Massachusetts Institute of Technology	15/1/24 08:03	2.3579
Ahmet Bahadır	Trabzon	Turkey	Boğaziçi University	17/1/24 21:11	1.683
Vadim	Ionas	Moldova	Denmark Technical University	20/1/24 12:47	1.6105
YUAN	FEI	Hong Kong	The Hong Kong Polytechnic University	21/1/24 15:09	1.3177
SIfat Ahmed	Tushar	Bangladesh	University of Dhaka	26/1/24 22:33	1.21
Karlo	Maksimović	Croatia	University of Zagreb	23/1/24 17:13	1.1979
Elias Koschier	Michail Ognyanov	Austria	–	4/2/24 12:40	0.9
Kai-Hsiang	Tu	Taiwan	National Taiwan University	4/2/24 11:01	0.9
Michał	Kołodziejczyk	Poland	Warsaw University of Technology	4/2/24 12:08	0.81
Marinus	Lehmann	Germany	University of Potsdam	28/1/24 16:42	0.8019
Nick	Hartmann	Germany	RWTH Aachen	11/2/24 12:19	0.72
Kerim	Cherkezov	Turkmenistan	Magtymguly Turkmen State University	15/2/24 10:23	0.7
Musa	Artykov	Turkmenistan	Magtymguly Turkmen State University	18/2/24 01:36	0.7
Dhruv	Shah	India	Indian Institute of Science	27/2/24 20:27	0.5
Dženan	Midžić	Bosnia and Herzegovina	University of Rijeka	10/3/24 11:39	0.5
Larrance	Xing	United States	University of Michigan Ann Arbor	26/2/24 02:10	0.5
Bayram Alp	Sahin	Turkey	Bogazici University	3/3/24 09:31	0.5
Md. Badhan	Mridha	Bangladesh	University of Dhaka	2/3/2024 15:39	0.5
Ricardo	Ochel	Germany	Universität Heidelberg	31/3/24 04:19	0.475
Konstantin	Tripunovski	North Macedonia	University of Ljubljana – Faculty of Medicine	3/4/24 15:18	0.3679

The list of countries:
Algeria, Argentina, Armenia, Australia, Austria, Azerbaijan, Bahrain, Bangladesh, Belarus, Belgium, Bosnia and Herzegovina, Brazil, Bulgaria, Cambodia, Cameroon, Canada, China, Croatia, Cuba, Czech Republic, Ecuador, Egypt, El Salvador, Estonia, Finland, France, Georgia, Germany, Greece, Guatemala, Honduras, Hong Kong, Hungary, India, Indonesia, Iran, Israel, Italy, Japan, Kazakhstan, Kenya, Kosovo, Kyrgyzstan, Latvia, Lithuania, Macau, Malaysia, Mexico, Moldova, Montenegro, Nepal, Netherlands, Nigeria, North Korea, North Macedonia, Pakistan, Peru, Philippines, Poland, Romania, Russia, Rwanda, Saudi Arabia, Serbia, Singapore, Slovakia, Slovenia, South Africa, South Korea, Spain, Sri Lanka, Suriname, Sweden, Switzerland, Syria, Taiwan, Tajikistan, Thailand, Turkey, Turkmenistan, Ukraine, United Arab Emirates, United Kingdom, United States, Uzbekistan, Vietnam.