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## electric field

### 6. Series 29. Year - 5. Particle race

Two particles, an electron with mass $m_{e}=9,1\cdot 10^{-31}\;\mathrm{kg}$ and charge $-e=-1,6\cdot 10^{-19}C$ and an alpha particle with mass $m_{He}=6,6\cdot 10^{-27}\;\mathrm{kg}$ and charge 2$e$, are following a circular trajectory in the $xy$ plane in a homogeneous magnetic field $\textbf{B}=(0,0,B_{0})$, $B_{0}=5\cdot 10^{-5}T$. The radius of the orbit of the electron is $r_{e}=2\;\mathrm{cm}$ and the radius of the orbit of the alpha particle is $r_{He}=200\;\mathrm{m}$. Suddenly, a small homogeneous electric field $\textbf{E}=(0,0,E_{0})$, $E_{0}=5\cdot 10^{-5}V\cdot \;\mathrm{m}^{-1}$ is introduced. Determine the length of trajectories of these particles during in the time $t=1\;\mathrm{s}$ after the electric field comes into action. Assume that the particles are far enough from each other and that they don't emit any radiation.

### 4. Series 29. Year - 2. Brain in a microwave

How far from a base transceiver station (BTS) do a person have to be, for the emission to be fully comparable with that of the mobile phone just next to somebody's head. Expect the BTS to broadcast uniformly into a half-space with the emission power 400 W. The emission power of a mobile phone is 1 W.

### 2. Series 27. Year - 2. Flying wood

We have a wooden sphere at a height of $h=1\;\mathrm{m}$ above the surface of the Earth which has a perimeter of $R_{Z}=6378\;\mathrm{km}$ and a weight of $M_{Z}=5.97\cdot 10^{24}\;\mathrm{kg}$. The sphere has a perimeter of $r=1\;\mathrm{cm}$ and is made of a wood which has the density of $ρ=550\;\mathrm{kg}\cdot \mathrm{m}^{-3}$. Assume that the Earth has an electric charge of $Q=5C$. What is the charge $q$ that the sphere has to have float above the surface of the Earth? How does this result depend on the height $h?$

Karel přemýšlel, co zadat jednoduchého.

### 1. Series 27. Year - 5. a bead

A small bead of mass $m$ and charge $q$ is free to move in a horizontal tube. The tube is placed in between two spheres with charges $Q=-q$. The spheres are separated by a distance 2$a$. What is the frequency of small oscillations around the equilibrium point of the bead? You can neglect any friction in the tube.

**Hint:** When the bead is only slightly displaced, the force acting on it changes negligibly.

Radomír was rolling in a pipe.

### 1. Series 27. Year - P. speed of light

What would be the world like if the speed of light was only $c=1000\;\mathrm{km}\cdot h^{-1}$ while all the other fundamental constants stayed unchanged? What would be the impact on life on Earth? Would it even be possible for people to exist in such a world?

Karel came up with an unsolvable problem.

### 4. Series 26. Year - E. Fun with straws

You can charge a regular plastic straw by rubbing it with a piece of fabric. This charge can be so large that the straw might even attach itself to a wall or a whiteboard. Explain this phenomena and estimate the charge you can put on a single straw.

**Hint:** You might need to use two straws.

Karel ran out of drawing pins.

### 3. Series 26. Year - S. tokamak

- Calculate the specific resistance of hydrogen plasma at temperature 1 keV. Compare your result with the resistance of common conductors.

- Calculate the current necessary to create a sufficiently strong poloidal magnetic field in a tokamak with a major radius of 0.5 m. The toroidal field is created using a toroidal coil with 20 windings per meter. The current inside this coil is 40 kA. The magnitude of the poloidal field should be approximately 1/10 of the magnitude of the toroidal field.

- Create a physical model of the field lines of the force field inside the tokamak, take a photo of it, and send it to us.

### 5. Series 25. Year - 2. electric equilibrium

An insulating rod of length $d$ and negligible mass can rotate around its middle point (see picture). There are small balls of negligible mass and charges $Q_{1}$ and 2$Q_{1}$ on both of its ends. Due to the weight $G$ (see picture) the system is in mechanical equilibrium. Distance $h$ under each of the small balls is a another ball with charge $Q$.

- What is the distance $x$ for which the rod is horizontal and is in equilibrium?
- What is the distance $h$ such that the rod is in equilibrium but there is no force on the pivot that holds it?

Dominika dug into old problem sets.

### 3. Series 25. Year - 5. gas leakage

Imagine two infinitely large grounded conducting planes distance $l$ apart. There is a charge placed between them and distance $x$ from one of them. What is the induced charge on the other plane?

Aleše napadlo při úniku.

### 1. Series 25. Year - 4. drrrrr

A small conductive ball of negligible size is bouncing between two charged plates of opposite polarities. What is the frequency of the resulting periodic motion of the ball? Voltage between the plates is $U$. When the ball touches a charged plate it charges to a charge of magnitude $Q$ whose polarity is the same as polarity of the plate. The ratio of kinetic energies of the ball before and after an impact is $k$.

**Bonus:** Does the output power of this resistor correspond to the energy losses during impacts?

Jáchym