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## electric field

### 5. Series 24. Year - P. chordates

As you know the amount of positive charges in the Universe is the same as the amount of negative charges. An argument for this statement is that if it were not so the repulsive electric force would be greater than the attractive gravitational force and objects would not hold together. However is this equilibrium perfect? What if there are more positive charges than negative ones and the electric force just reduces the gravitational attraction? Describe an experiment that would enable us to find out any discrepancy from this equilibrium. What is the minimum discrepancy we would be able to measure with such an experiment? As a measure of this discrepancy you can take the ratio of total charge/(positive charge-negative charge) in some large volume.

### 6. Series 22. Year - 2. escape from the charged sphere

Inside of grounded conductive sphere is made a small orifice, just big enough for a small charged particle to pass through. Assume the particle is placed in distance $d$ from the center of the sphere (see picture).

The charged particle is set free. How far it will fly off from the sphere? Try to use method of mirror potential.

### 5. Series 22. Year - S. games with electrons

• The second method to measure specific electron charge used by J. J. Thomson is observation of deflection of cathode ray by electric field. Assume apparatus as on the figure. How depends deflection of the beam observed on the screen on the right on the electrical voltage, speed and geometrical dimensions of apparatus?
• The one of problems which was J. J. Thomson facing during measurement of specific electron charge was following: After the beam entered magnetic field, the beam has spread into a bigger area (see figure). The dispersion is causing some error in measuring position of electrons (Thomson mentioned up to 20%). How can be this dispersion explained? How can be this inaccuracy improved (this is for bonus point)?
• By using the data from table calculate charge of electron in case, that the oil had density of 920 kg\cdot m^{ − 3}, air density was 1,2 kg\cdot m^{ − 3} and viscosity 17,1 \cdot 10^{−7} Pa\cdot s. Used electric field was 250 kV\cdot m^{ − 1}.

Zadali Pavel M. a Jakub B.

### 3. Series 21. Year - 4. particle in field

Assume constant electrostatic field in time. Lets insert a charged particle into the field with zero speed. When we record the trajectory of particle, we see that it does not depend on its mass. Can you explain it?

Na problém narazil Marek Scholz při programování zápočťáku.

### 2. Series 21. Year - 4. charged aerial

Two identical charges are at the end of stiff non-conductive rod. What power will be needed to rotate rod at constant angular speed with axis going through the middle of the rod? The friction is negligible

Úlohu vymyslel Martin Výška.

### 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.

### 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. 