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## wave optics

### (12 points)3. Series 35. Year - E.

Measure the rotation of the plane of polarization as a function of the sugar concentration in the solution.

### (10 points)2. Series 35. Year - S. compressing

What energy must a laser impulse lasting $10 \mathrm{ns}$ have in order for the shock wave generated by it to be able to heat the plasma to a temperature at which a thermonuclear fusion reaction can occur? What will be the density of the compressed fuel? **Note:** Assume that the initial plasma is a monatomic ideal gas.

### (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.

### (10 points)5. Series 34. Year - S. resonance and damped oscillations

- On a tense rope, waves can exist with the deflection $\f {u}{x, t}$ from the equilibrium, that satisfy the wave equation with damping

\[\begin{equation*}
\ppder {u}{t} = v^2 \ppder {u}{x} + \Gamma \pder {u}{x} ,
\end {equation*}\]
where $v$ is the phase velocity and $\Gamma $ is the coefficient of damping. Do a fourier substitution and find the dispersion relation. Solve it for the wavenumber $k$. What condition, in terms of frequency $\omega $, phase velocity $v$ and the coefficient $\Gamma $, must the waves meet in order to create nodes on the rope (i.e. points in which the rope stays in equilibrium position, but around which the rope is moving)?

- Consider a jump rope attached firmly at one end to a fixed wall. At the distance $L$ from the wall, we start moving the rope up and down to create waves. The jump rope has a linear density $\lambda $ and the constant tension $T$ in the direction away from the wall. The deflection then satisfies the equation

\[\begin{equation*} \ppder {u}{t} = \frac {T}{\lambda } \ppder {u}{x} . \end {equation*}\] For the deflection of the end of the rope that is moving satisfies $\f {u_0}{t} = A \f {\cos }{\omega _0 t}$. Assume the solution can be written in the form of two planar waves moving in the opposite direction to each other. Find the solution using only the parameters given in this problem statement, that is $T$, $\lambda $, $L$, $A$ and $\omega _0$. For certain frequencies, the solution has a diverging amplitude (i.e. growing beyond any limits). Find their values and the respective wavelenghts.

Štěpán was playing with a jump rope.

### (5 points)4. Series 34. Year - 3. curved optics

Let's have a point source of light and a planar glass panel with a refractive index $n = 1,50$. In the foot of the perpendicular from the source to the panel there are wavefronts with a radius of curvature $R = 5,00 \mathrm{m}$ inside the glass. What is the real distance between the source and the panel?

### (8 points)1. Series 34. Year - 4. solar sail

A solar sail with the surface area of $S = 500 \mathrm{m^2}$ and area density $\sigma =1,4 \mathrm{kg\cdot m^{-2}}$ is located at the distance of $0,8 \mathrm{au}$ from the Sun. What force does the solar radiation act on the sail at the beginning of the sail's motion? What is the acceleration of the sail at that moment? The luminosity of the Sun is $L_{\odot } =3,826 \cdot 10^{26} \mathrm{W}$. Assume that the radiation approaches the sail from a perpendicular direction and scatters elastically. **Hint:** We recommend you find the acceleration for small initial velocity $v_0$ and then let $v_0 = 0$.

Danka wants to fly.

### (9 points)5. Series 33. Year - 5. optically relativistic

Calculate the phase shift $\Delta \Phi $ when an optical beam with a wavelength $\lambda _0$ goes through a glass plate with thickness $h$ and the index of refraction $n$ that is moving along the beam with constant speed $v$ relative to a case when the plate is stationary. We are interested mainly about the first nonzero term of Taylor series of $\Delta \Phi (v)$.

Dodo at optic lab.

### (7 points)2. Series 33. Year - 4. butterflies

A rainbow-like, turquoise colour of the genus Morpho butterflies' wings' surface is a consequence of constructive interference of light reflected from layers of the transparent cuticle (cell layer on the wings' surface). Layers of thickness $h\_t = 63{,}5 \mathrm{nm}$ and refractive index $n\_t = 1{,}53$ are separated by $h\_a = 120{,}3 \mathrm{nm}$ thick air gaps (see figure). Estimate the wavelengths of visible light corresponding to interference maxima.

Domča was catching butterflies during the exam period in January.

### (6 points)1. Series 33. Year - 3. infra sauna

Dano continues with equiping of his mansion with another sauna—this time an infra sauna. He wants to place a tube lamp right underneath the ceiling of the sauna which is $H=2,5 \mathrm{m}$ above the ground. Suppose the source of radiation emits energy with the power per unit length of $p = 1,2 \mathrm{kW\cdot m^{-1}}$, a radiation of what intensity and total energy would reach the skin of a person situated approximately $h=50 \mathrm{cm}$ above ground? The lamp is a straight tube, shines in a homogeneous manner and reaches from wall to wall just under the middle of the ceiling.

**Hint:** For simplicity, approximate the sauna to be a room where the sides touching the lamp and the ceiling are mirrors and the other two sides and the floor absorb the light without remitting it back into the room.

Karel visited wellness in Slovakia.

### (9 points)1. Series 33. Year - 5. generally relativistic

Before he set off on his flight towards Mars, the Starman in his Tesla Roadster arranged with Musk that once he reaches the distance $r=5 \cdot 10^{6} \mathrm{km}$ from the centre of mass of the Earth, Musk will shine a powerful green laser at him. The wavelength of the laser increases under the influence of the gravitational field of Earth. Compare this change of the wavelength to the electromagnetic Doppler effect. Study each of these effects separately. Assume that the Starman is moving away from Earth with velocity $v=4 \mathrm{km\cdot s^{-1}}$.

Vašek wants to go on a trip with Starman.