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## electric field

### (11 points)3. Series 34. Year - P. wavy electromagnetism

What if the laws of nature weren't the same throughout the whole universe? What if they somehow changed with location? Let's focus on electromagnetic interaction. What would be the minimal change of the Coulomb's law constant as a function of distance, such that we could observe a deviation? How would we observe it?

### (10 points)2. Series 34. Year - 5. magnetic non-stationarities detector

The electrical circuit shown in the figure can serve as a non-stationary magnetic field detector. It consists of nine edges of a cube formed by electric wire. The electrical resistance of one edge is $R$. If this construction lies in a non-stationary homogeneous magnetic field, which has, for simplicity, a constant direction, and its magnitude changes slowly, then there are currents $I_1, I_2, I_3$ flowing at the marked spots. With the knowledge of these currents, determine the direction of the magnetic field in space and also the dependence of its magnitude on time.

Vašek thought that an electromagnetic induction problem would be welcome.

### (10 points)5. Series 33. Year - S. min and max

*We are sorry. This type of task is not translated to English.*

They had to wait a lot for Karel.

### (10 points)4. Series 33. Year - S.

*We are sorry. This type of task is not translated to English.*

### (8 points)5. Series 32. Year - 4. splash

Consider a free water droplet with radius of $R$. We start to charge the drop slowly. Find the magnitude of the charge $Q$ the drop needs to splash.

### (6 points)4. Series 32. Year - 3. levitating

Matěj likes levitating things and therefore he bought an infinite non-conductive charged horizontal plane with the charge surface density $\sigma $. Then he placed a small ball with given mass $m$ and charge $q$ above the plane. For which values of $\sigma $ will the ball levitate above the plane? What is the corresponding height $h$? Assume that the gravitational acceleration $g$ is constant.

Matěj would love to have levitation superability.

### (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.