(13 points)3. Series 36. Year - E. game of discharges

Charge the object by rubbing it and then measure the dependence of its self-discharge on time. Determine the electrical conductivity of air. Consider that the magnitude of the charge varies according to
\[\begin{equation*}
Q = Q_0 \eu ^{-\frac {\sigma }{\epsilon }t} ,
\end {equation*}\]
where $Q_0$ is the initial charge, $\epsilon $ is the permeability of the air, and $\sigma $ is the conductivity we are looking for. Hint: Hang two small metallic objects (e.g. nuts) at the same height on thin, long filaments. Then take a straw, rub it to charge it, and transfer some of the charges to the objects. They should begin to repel away from the straw. Afterwards, you can determine the product of the charges and the conductivity from their relative distances.

How far from the surface of the target (suppose it is made of carbon and the laser has wavelength of $351 \mathrm{nm}$) is critical surface situated and how far does two-plasmon decay occur, if the characteristic length of plasma^{1)}

^{1)}

The density of plasma $n_e$ is typically expressed as a funciton $n_e = f\(\frac {x}{x_c}\)$, where $x$ is the distance from the target and $x_c$ is so called characteristic length of plasma, which represents scale parameter for the distance from the target.))is~$50 \mathrm{\micro m}$? Next assume

that the density of the plasma decreases exponentially with distance from the target,

that the density of the plasma decreases linearly with distance from the target.

What energy must electorns have in order to go through the critical surface to the real surface of the target? To calculate the distance electron travels in carbon plasma use an empirical relationship $R = 0{,}933~4 E^{1{,}756~7}$, where $E$ has units of \jd {MeV} and $R$ has units of \jd {g.cm^{-2}}.

What is the distance that an electron has to travel in the electric field of the plasma wave in order to reach the energies determined in second exercise?

Which wavelengths of scattered light are present in the case of stimulated Raman scaterring for laser with wavelength of $351 \mathrm{nm}$?

(8 points)1. Series 35. Year - 5. mechanically (un)stable capacitor

Assume a charged parallel-plate capacitor in a horizontal position. One of its plates is fixed and the other levitates directly below it in an equilibrium position. The lower plate is not mechanically fixed in its place. What is the capacitance of the capacitor depending on the voltage applied? Is the capacitor mechanically stable?

(10 points)6. Series 34. Year - P. more dangerous corona

When there is a coronal mass ejection from the Sun, the mass will start to propagate with high velocity through the space. Sometimes the mass can hit the Earth and affect its magnetic field. Estimate the magnitude of the electric currents in the electric power transmission network on Earth which could be generated by such ejection. What parameters does it depend on? Comment on what effects would such event have on the civilisation.

Karel was at a conference and then he saw a video on the same topic.

(3 points)5. Series 34. Year - 1. the charge of the Earth

Find the total electric charge, that the Earth would need to let all electrons close to its surface fly away. How would this charge differ if it had to deflect protons?

(11 points)3. Series 34. Year - P. wavy electromagnetism

What if the laws of nature weren't the same throughout the whole universe? What if they somehow changed with location? Let's focus on electromagnetic interaction. What would be the minimal change of the Coulomb's law constant as a function of distance, such that we could observe a deviation? How would we observe it?

(10 points)2. Series 34. Year - 5. magnetic non-stationarities detector

The electrical circuit shown in the figure can serve as a non-stationary magnetic field detector. It consists of nine edges of a cube formed by electric wire. The electrical resistance of one edge is $R$. If this construction lies in a non-stationary homogeneous magnetic field, which has, for simplicity, a constant direction, and its magnitude changes slowly, then there are currents $I_1$, $I_2$, $I_3$ flowing at the marked spots. With the knowledge of these currents, determine the direction of the magnetic field in space and also the dependence of its magnitude on time.