A1  ^2.50 A narrow straight channel passes through the center of a fixed cube with a side \(a\). The cube is uniformly charged, the charge density is \(\rho\). The distance from the cube center to the point of intersection of the channel and a face is \(L\). In the channel there is a particle of a mass \(m\) and a charge \(q\). Find the period of small oscillations of the particle near the center. The gravitational interaction of the particle and the cube can be neglected. The cube and the particle are oppositely charged.

1 M1 The idea of the cube decomposition is stated: either into 2 cubes ($+\rho$, $-\rho$) or into a cube and 3 plates.	0.20
2 M1 The thickness of the plates is correctly related to the displacement of the particle $h_x$ (either $2h_x$ on one side, or $h_x$ on both sides and with opposite charges)	0.20
3 M1 A correct initial expression (e.g. in terms of an integral) for the electric field (or potential) induced by one of the plates at the center of the cube is given.	0.30
4 M1 The above equation is transformed such that it becomes independent of $a$ and the orientation of the channel	0.30
5 M1 The resulting equation $E=\cfrac{\rho h}{6\epsilon_0}$ is obtained	0.30
6 M1 The vector sum of forces is written (or shown that the electrostatic force is aligned with the channel)	0.20
7 M1 The resulting equation for the force is correct (both modulus and direction) $\vec{F} = \cfrac{q\rho}{3\epsilon_0}\vec{r}$ (The direction is the key here, and must be clearly stated)	0.30
8 M2 A differential of the electrostatic field (or potential) induced by an infinitesimal volume element is given	0.20
9 M2 A correct volume integral for the field (or potential) is given	0.20
10 M2 The above integral is transformed such that it becomes independent of $a$ and the orientation of the channel	0.60
11 M2 The resulting equation $E=\cfrac{\rho h}{6\epsilon_0}$ is obtained	0.50
12 M2 The resulting equation for the force is correct (both modulus and direction) $\vec{F} = \cfrac{q\rho}{3\epsilon_0}\vec{r}$ (The direction is the key here, and must be clearly stated)	0.30
13 M3 The proof of centrality of the electrostatic field (e.g. $V = Ax^2 + By^2 + Cz^2$)	0.50
14 M3 The proof of central symmetry of the electrostatic field (e.g. $A=B=C, V=A\cdot r^2$)	0.50
15 M3 The correct coefficient $A$ for the proof above is found (e.g. from Gauss law)	0.20
16 M3 Gauss law is written and correctly applied.	0.30
17 M3 The resulting equation for the force is correct $\vec{F} = \cfrac{q\rho}{3\epsilon_0}\vec{r}$	0.30
18 Equation of motion of an harmonic oscillator is obtained	0.20
19 The resulting equation for the period is correct (up to the signs of charges): $T = 2\pi \sqrt{\cfrac{3\epsilon_0 m}{|q||\rho|}}$ (Marked only if method is correct and justified)	0.40
20 The opposite signs of the charges are considered, and, as a result, period of oscillations contains $\sqrt{-\rho q}$	0.10
21 Arithmetic error	-0.10
22 Not used	-0.30