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## magnetic 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?

Karel was watching YouTube too much.

### (10 points)3. Series 34. Year - S. electron in field

Consider a particle with charge $q$ and mass $m$, fixed to a spring with spring constant $k$. The other end of the spring is fixed at a single point. Assume that the particle only moves in a single plane. The whole system exists in a magnetic field of magnitude $B _ 0$, which is perpendicular to the plane of movement of the particle. We will try to describe possible modes of oscillation of the particle. Start by the determination of equations of motion - do not forget to include the influence of the magnetic field.

Next assume that the particle oscillates in both of the cartesian coordinates of the particle and carry out Fourier substitution - substitute derivatives by factors of $i \omega $, where $\omega $ is the frequency of the oscillations. Solve the resultant set of equations in order to determine the ration of the amplitudes of oscillations in both coordinates and the frequency of oscillations. The solution obtained in this way is quite complicated, and better physical insight can be gained in a simpler case. From now on, assume that the magnetic field is very strong, i.e. $\frac {q ^ 2 B _ 0 ^ 2}{m ^ 2} \gg \frac {k}{m}$. Determine the approximate value(s) of $\omega $ in this case, always up to the first non-zero order. Next, sketch the motion of the particle in the direct (i.e. real) space in this (strong field) case.

Štěpán wanted to create a classical diamagnet.

### (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.

### (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.