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Magic field of Discworld is so strong that the speed of light does no longer have its common meaning. This applies only close to the surface, where the refractive index of the magic field has magnitude $n_0 = 2,00 \cdot 10^{6}$. The refractive index decreases with height $h$ as $n(h) = n_0\eu ^{-kh}$, where $k = 1,00 \cdot 10^{-7} \mathrm{m^{-1}}$. Calculate the optimal angle (measured from vertical direction) under which a light signal shall be emitted from one end of the Discworld to reach the opposite end in the shortest time possible. Diameter of the Discworld is $d = 15\;000 \mathrm{km}$ and speed of light in vacuum is $c = 3,00 \cdot 10^{8} \mathrm{m\cdot s^{-1}}$.

The FYKOS bird found an optical bench at the Faculty of Physics. The bench allows him to place different tools along an optical axis. He started to play with it and gradually placed onto it: a point source of light, a first lens, a second lens and a screen, with the same spacing between them (so the distance between the screen and the light source is three times bigger than any distance of two neighbouring tools). A sharp image of the source was created on the screen. Then, he dipped the whole system into an unknown liquid, which he found in a strange container. To his amazement, the image on the screen stayed sharp. Figure out the refractive index of the given liquid, which is certainly different from the refractive index of air. You can assume that the refractive index of air is unitary. One of the lenses has ten times bigger focal length than the other and both are thin, manufactured from a material with refractive index $2$.

We illuminate a mirror at an angle of $\alpha = 15\mathrm{\dg }$ with respect to the normal. We want the light to travel directly back to the source. For doing so, we can use a glass prism with an index of refraction $n = 1,8$. Find the angle $\eta $ as a function of $\alpha $ and $n$ (see the figure). The prism is placed into the air with an index of refraction $n_0$.

Hint: \[\begin{align*} \sin \(x + y\) &= \sin x \cos y + \cos x \sin y  , \\
\cos \(x + y\) &= \cos x \cos y - \sin x \sin y  , \\
\sin x + \sin y &= 2\sin \(\frac {x + y}{2}\)\cos \(\frac {x - y}{2}\)  , \\
\cos x + \cos y &= 2\cos \(\frac {x + y}{2}\)\cos \(\frac {x - y}{2}\)  . \end {align*}\]

Danka won the annual Derivative Bee and she obtained a statuette made of transparent material as a reward. This statuette is made in shape of a cube prism with an edge of $a = 5$ cm and height of $h \leq a$. No matter what angle she looks at the prism, she can only see the reflection on the side walls but not through it. What is the index of refraction of the material? The prism is placed in air.

A light ray falls on a glass plate with an absolute reflective index $n = 1,5$. Determine its angle of incidence $\alpha _1$ if the reflected ray forms an angle $60 \dg$ with the refracted ray. The board is stored in the air.

The solar constant, or more accurately the solar irradiance, is the influx of energy coming from the Sun at the distance where Earth is. It technically doesn't have a constant value, but let's suppose it is approximately $P = 1{,}370\,\mathrm{W\cdot m^{-2}}$. Also, suppose that Earth's orbit is circular and its axis of rotation is tilted with respect to the normal of the orbital plane by $23.5\dg $. What would be the maximum power captured by a solar panel of area $S= 1\,\mathrm{m^2}$ at the summer and winter solstice, if the panel lies flat on the ground in Prague (latitude $50\dg $ N)? Ignore the effects of any obstructions or the atmosphere.

What fraction of a spherical planet's surface cannot be seen from the stationary orbit above the planet? (A stationary orbit is one where the satellite stays fixed above a certain point on the planet.) The density of the planet is $\rho $ and its rotation period is $T$.

Consider an optical switch (transfer speed $10 \mathrm{Gb s^{-1}}$), whose output (after any necessary amplification) is used to illuminate the Moon. Thanks to the mirrors left behind by the Apollo mission, the signal comes back and can be used (after any necessary amplification) as an input to the switch. If we make sure the switch works reliably the transmitted data will circle in the system indefinitely. Thus we acquire a memory. What is its maximum capacity? Ignore any delays caused by the processing of the signal and any headers of the data.

Describe the imaging system of a microscope (consisting of two convex lenses) and that of a Keplerian telescope. Explain the difference in function and construction of a microscope and a telescope and sketch the rays passing through the systems. How can we usefully define magnification for these optical systems? Derive the equations for magnification.

Have you ever seen such a weird atmosphere? Up to a certain height the speed of light inside it is constant, $v_{0}$, but from that certain height the speed of light starts increasing linearly as $v(Δh)=v_{0}+kΔh$. At one point, exactly at the height where the speed of light starts changing, light beams are sent upward in all directions. Show that all these beams move along circular arcs and determine the radii of these arcs. Also find out the distance between the place where the the light was emitted and the point where the beams return to the original height.