**P. 4254.** A group of amateur astronomers observes a newly discovered binary star. According to their observations the two stars look like a pair of bright points, whose path is the ellipse shown in the figure. Ester, the most enthusiastic member of the group, recently learned Kepler's laws in the high-school. According to her knowledge the celestial bodies really move along elliptical paths, but the attractive centre is not at the symmetry centre *O* of the ellipse but at one of its foci. She suspects that the two stars revolve about an invisible black hole, which has much greater mass than that of the stars, and which is at the focus of the ellipse. Can Ester be right? How would you explain the observation?

(3 points)

**P. 4256.** A rubber cord is stretched out between two walls. At the point of the cord *C* a force of *F *along the direction of the cord is exerted.

*a*) What is the displacement of point *C* if the cord can be characterized with a spring constant *D*?

*b*) At which point of the cord must the force *F* be exerted when the displacement of *C* is the greatest?

*Data:* *F*=5 N, *a*=40 cm, *b*=10 cm, .

(4 points)

**P. 4262.** An electron is shot at an initial speed of *v*_{0}=4^{.}10^{7} m/s into a parallel plate condenser halfway between the plates. The distance between the plates and the length of the plates are unknown, and a constant voltage of *U*=500 V is applied between the plates. If a uniform magnetic field of induction *B*=6.25^{.}10^{-5} Vs/m^{2} is also applied between the plates, in the appropriate direction, the electron goes through the plates without changing the direction of its motion, at a constant speed.

*a*) Determine the electric field between the plates of the parallel plate condenser.

*b*) Find the distance *d* between the plates.

*c*) If the magnetic field is ceased the speed of the electron increase by 6% when passes the condenser. What is the length of the plates *L*?

(4 points)

**P. 4263.** The rate of change of the current in a toroidal coil is 10 A/s. The number of turns of the coil is 2000, its cross section is 2 *cm*^{2}, and the radius of its central circle is 0.5 m. At what acceleration does a proton or an electron starts to move if it is initially at the centre of the toroidal coil?

*Hint:* The electric field generated by the varying magnetic field of the toroidal coil is similar to the magnetic field generated by a circular current.

(5 points)