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## electric field

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

### Uranium storage

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.

### Heating wire

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.

### Capacity of a cube

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.

### 5. Series 20. Year - E. left-handed world

Measure optical activity of glucose solution as it depends on glucose concentration. Optical activity is rotation of polarisation direction of polarised light when passing through the solution. It is directly proportional to the optical path through the solution and depends on wavelength. Try to find/invent/remember how we explain optical activity at molecular level.

Measurement of optical activity is used for measuring sugar concentration in solutions. Is this method reliable? Has each sugar the same optical activity?

Úloha napadla Honzu Prachaře při čtení Feynmanoých přednášek z fyziky.

### 2. Series 20. Year - 3. illumination of table

Find such placement of fluorescence tubes at the ceiling of study room, which is 3 m above the top surface of desk, that intensity of illumination will not vary more than 0,1 %.

Úloha napadla Honzu Prachaře při čteni Feynmanoých přednášek z fyziky.

### 6. Series 18. Year - E. catch a photon

Measure the speed of light in vacuum. Use any method, for example use microwave oven.

Co jiného dát jak exp do roku fyziky.

### 3. Series 18. Year - 3. charged cube

What is the ratio of the electrostatic potentials in vertex and in the centre of non-conductive homogeneously charged cube? The total charge is $Q$, the edge length is $a$. Assume the electrostatic in infinity to be zero.

Vymyslel Pavel A.