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## magnetic field

### (10 points)6. Series 37. Year - 5. oscillating magnets

Consider two identical dipole magnets, which we fix so that they can rotate in the same plane without friction (their axes of rotation are parallel). If we deflect the magnets slightly out of their equilibrium position, they begin to oscillate. Find the eigenmodes of these oscillations and calculate their frequencies. Discuss what the motion of the magnets will be like for general initial deflection (you don't have to explicitly calculate this case). The magnets have a magnetic moment $m$, a moment of inertia about the axis of rotation $J$ and the mutual distance between their centers is $r$.

Jirka stole the problem from Výfuk.

### (10 points)5. Series 37. Year - S. we are spending electricity

- The aluminum smelter annually produces $160~000 t$ of aluminum, which is produced by electrolysis of alumina using a DC voltage of $U=4{,}3 \mathrm{V}$. Determine how many units of nuclear power plant with a net electrical output of $W_0=500 \mathrm{MW}$ are equivalent to the energy consumed by the aluminum smelter.
- In the second half of the last century, conventional electrical units were based on the values of the frequency of the cesium hyperfine transition $\nu \_{Cs}=9~192~631~770 Hz$, the von Klitzing constant $R\_K=25~812.807 \ohm $ and the Josephson constant $K_J=483~597.9e9 Wb^{-1}$. Determine the value of the coulomb $1 \mathrm{C}$ using these constants. {enumerate}

Dodo has dead batteries.

### (6 points)4. Series 37. Year - 3. step here, step there

Consider a homogeneous magnetic field of induction $B_1$, which spans a half-space bounded by the plane of interface $y=0$, beyond which is an equally oriented, also homogeneous magnetic field of induction $B_2$. An electron flies out of the plane perpendicularly to it and the field lines (as in the figure) with velocity $v$. Determine the size and the direction of its average velocity parallel to the plane of the interface.

*Bonus:* Consider now that the magnitude of the field changes linearly as $B = B_0 \(1+\alpha y\)$ and its direction is in the positive direction of the $z$-axis. Again, determine the magnitude and direction of the average velocity of the electron parallel to the interface plane. The electron is initially emitted as in the previous case.

Jarda is going one step forward and two steps back

### (8 points)5. Series 36. Year - 5. xenon was wandering

A once positively ionized xenon atom flew out from the center of a large cylindrical coil with velocity $v=7 \mathrm{m\cdot s^{-1}}$ and began to move through a homogeneous magnetic field, which is in a plane perpendicular to the magnetic lines of force. At a certain point the coil is disconnect from the source, thus its induction begins to decrease exponentially according to the following equation $\f {B}{t}=B_0\eu ^{-\Omega t}$, in which $B_0=1,1 \cdot 10^{-4} \mathrm{T}$ and $\Omega =600 \mathrm{s^{-1}}$. What is the deviation from the initial direction after the atom is stabilized?

Vojta spent several hours thinking about a reasonable problem assignment with a clever solution, but ultimately, it ended horrendously. And he has yet to see the solution.

### (10 points)2. Series 36. Year - 5. magic magnetic stick

Consider a thin magnet placed in the middle of a thin hollow rod of length $l$. The material of the rod is capable of shielding the magnetic field. Just beyond the end of the rod, the magnetic field flux is equal to $\Phi $. Calculate the direction and strength of the magnetic field in a plane perpendicular to the rod passing through its center as a function of the distance $r$ from the rod.

Adam made a blowgun so that he could blow magnets at his classmates in lectures.

### (8 points)5. Series 35. Year - 5. alternating triangle

Let us construct the finite Sierpiński triangle of a degree $N$ (for $N = 1$ it is a single triangle, in case of $N = 2$ it is four triangles, etc.). The bases of small triangles (that the Sierpiński triangle is made of) consist of a resistor with resistance $R = 150 \mathrm{\ohm}$, the left legs are coils of inductance $L = 0{,}4 \mathrm{H}$ and the other legs are capacitors of capacitance $C = 20 \mathrm{\micro F}$. We measure the impedance between the triangle's bottom left and right corners. The angular frequency of the source is $\omega = 50 \mathrm{s^{-1}}$. Find the recurrent relation for the measured impedance and find its value for $N = 7$. What does the recurrent formula looks like if we replace coils and capacitors with resistors $R$? Determine its numerical value for $N = 15$.

Honza likes fractals.

### (10 points)5. Series 35. Year - S. stabilizing

- What intensity must a laser with a wavelength of $351 \mathrm{nm}$ have in order to stabilize a Rayleigh-Taylor (RT) instability using the surface ablation of a fuel pellet? Suppose the boundary between the ablator and DT ice is corrugated with a wavelength of

- $0,2 \mathrm{\micro m}$,
- $5 \mathrm{\micro m}$.

- How will the intensity of the laser change if we also apply a magnetic field with magnitude $5 \mathrm{T}$?
- What else can help us minimize the RT instability?

### (5 points)3. Series 35. Year - 3. two solenoids

Consider two coils wound around a common paper roll. First coil has a winding density of $10 \mathrm{cm^{-1}}$ and the second coil has a winding density of $20 \mathrm{cm^{-1}}$. The paper roll is $40 \mathrm{cm}$ long and has $1 \mathrm{cm}$ in diameter. Both coils are wound along the whole length of the roll, with the second coil wound around the first one. Considering the dimensions of the roll, we can neglect the boundary effects and assume that the coils behave as perfect solenoids. Now consider connecting the coils in series. This configuration can be substituted by a circuit with a single coil. What is the inductance of the substituting coil?

Jindra played games with paper rolls.

### (10 points)6. Series 34. Year - P. more dangerous corona

When there is a coronal mass ejection from the Sun, the mass will start to propagate with high velocity through the space. Sometimes the mass can hit the Earth and affect its magnetic field. Estimate the magnitude of the electric currents in the electric power transmission network on Earth which could be generated by such ejection. What parameters does it depend on? Comment on what effects would such event have on the civilisation.

Karel was at a conference and then he saw a video on the same topic.

### (10 points)6. Series 34. Year - S. charged chord

Assume a charged chord with linear density $\rho $, uniformly charged with linear charge density $\lambda $. The tension in the chord is $T$. It is placed in a magnetic field of constant magnitude $B$ pointing in the direction of the chord in equilibrium. Your task is to describe several aspects of the chord's oscillations. First, we want to write the appropriate wave equation. Neglect the effects of electromagnetic induction (assume the chord to be a perfect insulator; that also means the charge density does not change) and find the Lorentz force acting on an unit length of the chord for small oscilations in both directions perpendicular to the equilibrium position. Use this force to write the wave equation (which will also include the effects of the tension). Apply the Fourier substitution and determine the disperse relation in the approximation of a weak field $B$; more specifically, neglect the terms that are of higher than linear order in $\beta = \frac {\lambda B}{k \sqrt {\rho T}} \ll 1$, where $k$ is the wavenumber. Find two polarization vectors, this time neglect even the linear order of $\beta $. Now suppose that in a particular spot on the chord, we create a wave oscilating only in one specific direction. How far from the original spot will be the wave rotated by ninety degrees from the original direction?

Štěpán was nostalgically remembering the third serial task.