# 2021-P1 Best Solutions and Final Results

Problem 1 – Dumbbell in a magnetic field

By 18th February 2021, there are 976 registered participants from 71 countries. In total, 97 solutions were submitted for problem 1, out of which 31 were correct.

As usual, when selecting the best solutions, preference has been given to those solutions which have been submitted before the publication of the first hint. Overall, there were eight such solutions: from Vlad-Ștefan Oros, Oliwier Urbański, Zhening Li, Elio Moreau, Eddie Chen, Antonia Ghita, Joshua Teo, Nikita Semenin.

With this problem, it was easy to overlook the fact that in the initial value of the generalized momentum, the dominating contribution is given by the polarization of the balls. Missing this fact did not mean as much lacking in physics, but being slightly careless, and the formula would have been accurate enough if the magnetic field had been a few times weaker (similarly I have been overlooking all geometry-related mistakes by a small parameter R/L). Keeping in mind this, together with the fact that it was a really difficult problem, all submissions with this mistake had been considered as correct ones.

However, when selecting the best solutions, of course preference has been given to completely correct solutions. This leaves us with just two submissions, from Zhening Li and from  Eddie Chen. Out of those two, the solution of Zhening Li is more accurate as it further improves the accuracy by considering image charges induced on the spheres (just a small comment to his solution: while dipole radiation does carry away a small fraction of energy, it is very weak due to the smallness of the  acceleration; the main factor leading to the decay of oscillations is the Ohmic loss in the rod when charge is flowing from one ball to another). Furthermore, it is a very nicely written solution with detailed explanations, typeset in LaTeX. Therefore, Zhening Li will receive 60% of the bonus while  Eddie Chen will get 40%.

Apart from these two, there are two more solutions well worth publishing:
Vlad-Ștefan Oros has also nice detailed explanations and estimates the charge relaxation time which is an important parameter of the problem;
Nikita Semenin who has written and analyzed the full set of differential equations describing the intermediate dynamics of the system.

Final results for Problem 1

Pre-university students

University students

Participating countries: Algeria, Azerbaijan, Australia, Bangladesh, Belarus, Belgium, Bosnia and Herzegovina, Bolivia, Brazil, Bulgaria, Cambodia, Canada, China, Croatia, Czechia, Egypt, El Salvador, Estonia, Finland, Georgia, Germany, Greece, Hungary, India, Indonesia, Iran, Israel, Italy, Japan, Kazakhstan, Kenya, Korea, Kyrgyzstan, Latvia, Lebanon, Lithuania, Macedonia, Malaysia, Mexico, Moldova, Mongolia, Nepal, Nigeria, North Macedonia, Pakistan, Peru, Philippines, Poland, Romania, Russia, Saudi Arabia, Serbia, Singapore, Slovakia, Slovenia, South Africa, South Korea, Spain, Sri Lanka, Sweden, Switzerland, Syria, Taiwan, Thailand, Turkey, Turkmenistan, United Arab Emirates, United Kingdom, United States, Uzbekistan, Vietnam.