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## electric field

### (7 points)1. Series 32. Year - 3. unstable

We have 8 point charges (each of magnitude $q$) located on the vertices of a cube. Find out the value of a point charge $q_0$ that needs to be placed in the middle of the cube, so that all charges remain in balance. Is this equilibrium stable?

Matej wanted to pose a problem that even a professor couldn't work out.

### (5 points)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.

### (2 points)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 points)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.

### (5 points)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.

### (5 points)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.

### (8 points)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.

### (6 points)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.

### (2 points)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.

### (5 points)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.