Results after Problem 6

The list of the contest leaders after the sixth problem:

Points for Problem No 6:

Correct solutions (ordered according to the arrival time; best solutions in bold):

1. Szabó Attila (Hungary)

2. Nikita Sopenko (Russia)

3. Lars Dehlwes (Germany)

4. Kohei Kawabata (Japan)

5. Kai-Chi Huang (Taiwan)

6. Petar Tadic (Montenegro)

7. Bruno Bento Barros de Araújo (Brazil)

8. Ilie Popanu (Moldova)

9. Selver Pepić (Bosnia-Herzegovina)

10. Ion Toloaca (Moldova)

11, Nadezhda Vartanian (Russia)

12. José Luciano de Morais Neto (Brazil)

13. Jakub Šafin (Slovak)

14. Ivan Tadeu Ferreira Antunes Filho (Brazil)

15. Jôhanes Sebástian Paiva Melo (Brazil)

16.  Adrian Nugraha Utama (Indonesia)

17. Andrew Zhao (United States)

18. Ng Fei Chong (Malaysia)

19. Jakub Supeł (Poland)

20. Alexandra Vasileva (Russia)

21. Lorenzo Comoglio (Italy)

22. Cristian Zanoci (Moldova)

23. Dinis Cheian (Moldova)

(The list is ordered according to the arrival time)

Overall number of registered participants: 258 from 44 countries.

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The following hints were given for the last week. It is clear that the charges can be only on the surface of the metal: on the spherical part, and on the two adjacent surfaces of the plane cut (volume charges inside the metal would create an electric field therein). Note also that at each point of the cut, the charges at the two sides need to compensate each other (uncompensated charge would create an electric field in the metal near the cut). You need to figure out, how the charges need to be distributed on these surfaces to ensure that the field will be potential (whichever way an imaginary test charge moves from one piece of the sphere to another one, the work done by the electric field should remain the same). Also, the electric field created by the charges on the spherical surfaces should cancel out inside the region occupied by the metal.

The following methods can be useful: study the Gauss law for a small volume near the spherical surface to express the electric field outside the sphere via the surface charge density at that location; study the circulation theorem for a small contour near the surface (but outside the sphere): near the surface, the electric field is perpendicular to the surface (ie. radial), so that the tangential segments of the contour give no contribution to the circulation, which makes it possible to draw useful conclusions regarding the behaviour of the  radial  field (about the dependence on the tangential coordinate).