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### 5. Series 22. Year - 4. internet

Assume a straight optical fibre. The incoming light enters fibre at maximum angle $α$ from the axis of fibre (higher angles will not be guided by fibre). What is the minimum length of the light pulse to guarantee no overlap between 0 and 1 bit (e.g. at least for short interval the intensity of signal must be minimal or maximal). The length of fibre is $d$.

na schůzku donesl Honza Jelínek

### 4. Series 22. Year - 2. on a thin ice

It is known, that the ice subjected to higher pressure has lower melting temperature. This effect is used also during ice skating. Is this effect strong enough at extremely low temperatures (e.g. is pressure from skate big enough to melt the ice)? If not, what is causing sliding?

Při návštěvě kluziště si počítal Dan.

### 3. Series 22. Year - P. life at Titan

Titan (a satellite of Saturn) is very cold (surface temperature is approximately 94 K) and has bulky nitrogen atmosphere, icy surface with lakes made from hydrocarbons. Radar on Cassini probe orbiting Titan found, that objects on the surface rotate faster then the moon itself (approximately by 0$,36°\;\mathrm{year}^{-1})$. Scientific explanation is, that the icy layer on the surface of oceans is affected by a wind. The assumption about moon rotation is, that it is synchronised with orbiting around Saturn

Another hint for existence of under-surface ocean was given by Huygens probe, which landed on the moon surface. During the descent it measured strong electromagnetic waves at frequency approximately 36 Hz. Such waves are amplified at the interface between e.g. water and ice under the surface.

Suggest some methods, to confirm or disprove the existence of hidden ocean on Titan.

V aktuálním dění zaujalo Honzu P.

### 2. Series 22. Year - 2. find the secret of calliper

Explain, how calliper (vernier caliper) works and how it is possible to measure up to 1/10th mm if the main scale is only 1mm!

nad tajemstvími života se zamyslel Marek Scholz

### 1. Series 22. Year - 2. pirate and golden reward

One pirate should get 10 golden coins. However the captain does not want to give money so easily and melts the gold to a cylinder shape. And together prepares similar cylinder, made from brass with identical dimensions. Because the golden cylinder has inside some air, both weight the same. What is the best way to select the golden cylinder?

Úlohu vymyslel kolega Mirka Beláňe.

### 6. Series 21. Year - P. mission impossible

Make a plan how to liberate a Fykos-bid from custody. Do not forget to make plan B and C.

Vyplodil Honza Prachař.

### 5. Series 21. Year - E. milestones of life in Rama

Will have the Rama (brand name for margarine) another physical properties after you will melt it and let it solidify back? We suggest to measure density, viscosity and colour.

Vytlačil Marek Pechal.

### 2. Series 21. Year - E. bubo bubo

Verify experimentally following hypothesis: the rotation of Earth causes water on north hemisphere to swirl to right, on south hemisphere to left. For your conclusion to have relevance, enough number of measurements must be done at different conditions.

Napadlo zadat Honzu Prachaře.

### 2. Series 21. Year - S. cutting of wild plains

<h3>Uranium storage</h3>

Very important question is storing of radioactive waste. Usually it is stored in cylindrical containers immersed in water, which keeps the surface at constant temperature 20 °C. Your task is to find the temperature distribution inside containers of square base of edge length 20 cm. Container is relatively long, therefore just temperature distribution in horizontal cross section is of interest. Uranium will be in block of square base of edge 5 cm. From the experience with cylindrical capsules we know, that it will have constant temperature of about 200 °C.

<h3>Heating wire</h3>

Lets have a long wire of circular cross section and radius $r$ from a material of heat conductivity $λ$ and specific conductivity $σ$. Then a electric field is applied. Lets the electric field inside the wire is constant and parallel with the axis of the wire and the strength is $E$. Then the current through wire will be $j=σE$ and will create Joule's heat with volume wattage $p=σE$.

Because the material of the wire has non-zero temperature conductivity, some equilibrium gradient of temperature will form. The gradient fulfills Poisson's equation $λΔT=-p$. Assume, that the end of wire is kept at temperature $T_{0}$. This gives a border condition needed to solve the equation. Due to symmetry we can take into account only two dimensions: on cross section of wire (temperature will be independent of shift along the axis of symmetry). Now it is easy to solve the problem with methods described in text.

However, we will make our situation little bit more complex and will assume, that specific electrical conductivity $σ$ is function of temperature. So we will have a equation of type Δ$T=f(T)$.

Try to solve this equation numerically and solve it for some dependency of conductivity on temperature (find it on internet, in literature of just pick some nice function) and found temperature profile in wire profile. Try to change intensity of electric field $E$ and plot volt-amper characteristics, you can try more than one temperature dependency. $σ(T)$ (e.g. semiconductor which conductivity increase with temperature, or metal, where conductivity is decreasing) etc.

Do not limit your borders, we would be glad for any good idea.

<h3>Capacity of a cube</h3>

Calculate capacity of ideally conductive cube of edge length 2$a$ (2Ax2Ax2A). If you think, it is simple, try to calculate for cuboid (AxBxC) or other geometrical shapes.

**Hint:**
Capacity is a ration of the charge on the cube to the potential on the surface of cube (assuming that the potential in infinity is zero). Problem can be solved by selecting arbitrary potential of cube and solving Laplace equation Δ$φ=0$ outside of the cube and calculating total charge in cube using Gauss law. E.g. calculating intensity of electrical field and derivating potential and calculation of flow through nicely selected surface around the cube.

Final solution is finding a physical model, its numerical solution and realization on computer. More points you will get for deeper physical analysis and detailed commentar. For algorithm you can also get extra points.

Zadal spoluautor seriálu Lukáš Stříteský.