Problem Statement of Series 4, Year 38

Remote access is often used while observing with telescopes at the La Silla observatory in Chile. When measuring the data transfer speed, we usually get a time of $t=213\,\mathrm{ms}$ for the signal passage from Prague to the observatory and back. How could such a connection be realized? Consider these two cases: The data transfer works via geostationary telecommunication satellites or a fiber optic connection.

The long forgotten gods of old became very bored while observing Earth. They therefore decided to transform the spherical Earth into a cylinder. The cylinder's axis of rotation passes through the center of its base and is perpendicular to it. What will be the ratio of the day length on the new Earth to the original day length? While they are gods, they are not magicians. Thus, the mass, density, and angular momentum of the cylinder remain the same as those of the original Earth. The height of the cylinder equals the diameter of the original Earth.

A small sphere is positioned at the furthest possible distance from Earth such that that it remains completely within the planet's shadow. At what temperature will the sphere reach equilibrium, assuming Earth behaves as a black body with a homogeneous temperature of $T_{\mathrm{E}} = 20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$? Neglect all light sources other than the Sun and assume that light rays propagate in straight lines, ignoring refraction in the atmosphere or relativistic effects.

Jarda is preparing for his special relativity exam in his vacation house on one of Jupiter's moons. He was not keeping track of time and found out his exam begins in two hours (his time is synced with Earth). He got into his extremely fast rocket and set out for Earth. At the time of takeoff, the distance between his rocket and Earth equaled $8\,\mathrm{AU}$. He wants to study during his journey, but he's learning at a rate $1.5$ times slower compared to when he's sitting in front of the examination room, as he needs to focus on controlling the ship while in flight. At what speed does he need to be flying to learn as much as possible? The ship is flying at a constant speed; do not consider the time required to accelerate and decelerate.

Jarda is standing at a tram stop, waiting for the tram. However, it still hasn't arrived, so he decides to walk at a speed $v$ toward an information board located dd meters away to check the timetable. Next to the board, someone is passing the time by smoking a cigarette. Determine when Jarda will get close enough to the smoker to smell the smoke. The concentration of smoke particles at a distance $d_0 = 1\,\mathrm{m}$ from the smoker is $c_0$. Jarda will notice the smell if the concentration of smoke particles at his position reaches $c_0/N$. Consider the smoker to be a symmetrically spherical emitter of smoke and assume there is no wind.

Propose as many methods as possible for separating substances from a mixture and provide a detailed explanation of the physical principle and experimental procedure for at least three of them. Each one of the three methods described in detail should be based on a different physical or chemical property of the substances.

Find all-purpose flour and poppy seeds in the kitchen. Verify whether their angle of repose depends on the height of the pile formed by pouring. Additionally, measure the dependence of the angle of repose on the mass concentration of the mixture of plain flour and poppy seeds.

1. The geometric surface area of our platinum electrode is $4\,\mathrm{cm^2}$. However, its surface is very rough, so the active surface area may be higher. In an experiment, we measured the capacitance of the whole electrode to be $700\,\mathrm{\upmu{}F}$. If we estimate the distance of adsorbed ions in the solution from the platinum surface to be $1\,\mathrm{nm}$, how many times larger is the active surface area compared to the geometric area? The experiment takes place in water with $\epsilon_{\mathrm{r}} \doteq 80$. – 2 points

2. Draw the impedance spectrum in a Nyquist plot for a resistor $R = 23\,\mathrm{m\Omega}$, a capacitor with capacitance $C = 0.5\,\mathrm{mF}$, and CPE with parameters $Q = 0.3\,\mathrm{\Omega^{-1}\cdot s^{\alpha}}$ and $\alpha = 0.6$ for the frequencies ranging from $f_1 = 1\,\mathrm{kHz}$ to $f_2 = 10\,\mathrm{kHz}$. – 2 points

3. Determine all the parameters of a Randles circuit from the provided spectrum. The data points are distributed logarithmically over the frequency range from $10\,\mathrm{Hz}$ to $10\,\mathrm{kHz}$, with 5 data points per one frequency decade. – 3 points

   

4. Impedance spectra of a simple reaction, described by a Randles circuit, were measured at a DC current $I$ given in the table. From curve fitting the spectra, the ohmic resistance was found to be $R_\Omega = 55\,\mathrm{\Omega}$ for all measured values. The charge transfer resistance $R_{\mathrm{ct}}$ values are listed in the table below. Assume the measurements were conducted in the Tafel regime. Determine the parameter $b$ in the exponential form of the Tafel equation $j = j_0 \exp\!\left( \eta / b\right)$ derived in the third episode of the series. – 3 points

   Values of the current and the resistance.

   measuring	$\frac{I}{\mathrm{mA}}$	$\frac{R_\mathrm{ct}}{\mathrm{\Omega}}$
   1	0,13	208
   2	0,24	99
   3	0,57	45
   4	1,11	22
   5	2,04	14