**P. 4839.** A solenoid is made by winding tightly (in one layer) a piece of enamelled wire of resistivity \(\displaystyle \varrho\), and of radius \(\displaystyle r\ll R\) around an insulating cylinder of length \(\displaystyle \ell\) and of radius \(\displaystyle R\ll \ell\). The solenoid is connected to a voltage supply of voltage \(\displaystyle U\). To what power of the radius \(\displaystyle r\) are the following quantities proportional?

\(\displaystyle a)\) The current in the wire;

\(\displaystyle b)\) the self-inductance of the solenoid;

\(\displaystyle c)\) the magnitude of the magnetic induction at the geometric centre of the solenoid.

(4 points)

**P. 4840.** A thin rod of length \(\displaystyle L\) is moved in a vertical plane in a right corner such that the speed of the lower, horizontally moving end of the rod has a constant speed of \(\displaystyle v_0\), and the other end touches the vertical wall at every moment. At both ends of the rod there are two small charged spheres of charge \(\displaystyle Q/2\) (one ball at each end). The whole system is in uniform magnetic field of magnetic induction \(\displaystyle \boldsymbol B\). The magnetic induction is horizontal and perpendicular to the plane of the *figure* pointing into the paper.

\(\displaystyle a)\) What is the magnitude and the direction of the force exerted on the rod by the magnetic field, when the rod makes an angle of \(\displaystyle \alpha\) with the horizontal?

\(\displaystyle b)\) Determine this force in the same position of the rod, if there are no charged spheres at the ends of the rod, but there is only one charged sphere at the midpoint of the rod. The charge on this sphere is \(\displaystyle Q\).

(5 points)

**P. 4841.** In case of the Compton effect ``stationary'' electrons are ``bombarded'' by photons which photons have the same energy as the rest energy of electrons. Among the above described photons there will be some whose linear momentum will be equal in magnitude to the magnitude of the linear momentum of the electrons with which the photons collided. Considering these cases determine the

\(\displaystyle a)\) angle between the scattered photons and the electrons with which the photons collided;

\(\displaystyle b)\) The speed of the electrons with which the photons collided.

(5 points)

**P. 4842.** There are small holes on an opaque sheet, the holes positioned in a regular hexagonal grid as shown in the *figure.* The sheet is illuminated by monochromatic laser light of wavelength \(\displaystyle \lambda\) perpendicularly to the sheet.

What kind of diffraction pattern can be observed on the screen which is placed at a distance of \(\displaystyle L\) from the sheet, if the distance between the holes is \(\displaystyle d\)? What can be stated about the brightness of the peak intensities with respect to each other? (It can be assumed that \(\displaystyle L\ll d \ll \lambda\).)

(6 points)