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## magnetic field

### (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.

### (7 points)5. Series 33. Year - 4. a strange loop

A circular metal loop of mass $m = 18 \mathrm{g }$, radius $r = 15 \mathrm{cm}$ and electrical resistance $R = 3{,}5 \mathrm{m\Ohm }$ is at rest. By the resistance of a loop we mean resistance between the ends of a wire created by cutting the loop in one place. At the time $t = 0$ we create a homogenous magnetic field perpendicular to the plane of the loop. The magnetic field strength changes as a function of time $B(t) = \alpha t$, where $\alpha = 1 \mathrm{mT\cdot s^{-1}}$ is a constant. Because of the nonstationary magnetic field, the loop will start to turn slowly around it's axis. Calculate the angular velocity $\omega $ at time $t = 0{,}1 \mathrm{s}$. Neglect the deformation of the loop.

Vašek likes bizzare phenomena.

### (8 points)3. Series 32. Year - 4. destruction of a copper loop

A copper flexible circular loop of radius $r$ is placed in a uniform magnetic field $B$. The vector of magnetic induction is perpendicular to the plane determined by the loop. The maximal allowed tensile strength of the material is $\sigma _p$. The flux linkage of this circular loop is changing in time as $\Phi (t) = \Phi _0 + \alpha t,$ where $\alpha $ is a positive constant. How long does it take to reach $\sigma _p$?

**Hint:** Tension force can be calculated as $T = |BIr|$.}

Vítek thinks back to AP Physics.

### (12 points)3. Series 31. Year - E. magnetically attractive

You got a planar magnet (magnetic foil) together with the tasks of these series. This magnet is a bit different than a rod magnet. The south and north poles are alternating parallel lines. When approaching the ferromagnetic surface, a magnetic circuit is created which holds the magnet (for example, on the fridge) and can carry even a picture on itself. Your tasks are:

- Measure the area and thickness of the film which you be used for your experiments.
- Measure the mean distance between the two closest same magnetic poles (twice the distance of opposite poles).
- Measure the maximum payload (ie. weight without magnet weight) which can be carried by a $1 \mathrm{cm^2}$ of a magnet if the magnet load is even if the magnet is attached to the bottom of the horizontal plate. The plate should be approx. $1 \mathrm{mm}$ thick sheet made of magnetically soft steel.

Charles obtained a magnetic foil.

### (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.

### (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.

### (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. Series 24. Year - 1. Warm-Up

*Jakub's breakfast*

Every morning Jakub enjoys his favourite cereals which he pours into a bowl of milk. Assume the bowl is of circulur frustum shape with upper and lower radius $R$ and $r$ respectively ($R$ \geq r$)$, that the cereals are little solid spheres and that before he puts the cereals into the bowl there is milk of height $h$. What is the maximum amount of cereals he can fit into the bowl? You also know that the fraction of volume the cereals occupy inside a fully filled box is $\kappa$.

*Magnetic monopole*

Let's have a metal plate magnetized in such a way that the upper and lower sides are the north and south poles respectively. We use these plates to create two semispheres with the outer side being the north pole. Now, if we glue these two semispheres together, we effectively get a magnetic monopole, which, as we know, can not exist in our world. Where did we go wrong?

eee

### 5. Series 23. Year - 2. a spring of Lukáš’

Lukáš found an old sofa spring of force constant $k$, coil radius $r$, length $l$ and the number of coils $n$. Since he was bored, he connected the spring to electric current $I$. How did the action change the force constant of the spring?

Lukáš's idea after Aleš told him to have one