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## wave optics

### 4. Series 23. Year - S. Maxwell

*We are sorry. This type of task is not translated to English.*

### 3. Series 23. Year - 1. indistinguishable people on Earth

What is the maximum distance for two people to be indistinguishable to others whenever they are visible? Do not forget that people are point light sources at 2 meters height and the Earth is an ideal sphere.

jmi

### 3. Series 23. Year - 3. Hospodine, pomiluj ny! (medieval Czech song)

How grows the volume (define yourself) of choir with the number of its members? What is the conclusion? Members of the choir can be approximated as point sound sources of the same amplitude and frequency, but shifted by a random phase. All point singers are in one place.

jmi

### 3. Series 23. Year - S. game with shadows

- In serial we discussed discrete distribution of point sources on a line and its projection onto another line. Now assume, that the points are distributed on a plane and the screen is the plane parallel to it. Describe the distribution of intensity on the screen in case, that light sources:

- lie at one line with equal spacing $d$.

- are as two parallel lines, where the distance between two of them is $d$.

- lie in the corners of rectangle network, where rectangles have sides $a$, $b$.

- Assume following situation: before the screen, presented by a plane $xy$ is a disc of radius $R$, parallel to the plane. The screen is illuminated from the side of the disc by a beam of parallel beams perpendicular to the screen $xy$. Explain, why this situation can be described using point light sources located continuously in the same plane as disc is located, excluding the disc itself. Find the intensity distribution in plane $xy$ as function of $x$ and $y$ (you can supply just integral, not it solution) and show, that point opposite the center of disc shows strange behaviour, which we would not expect from ray optics

### 2. Series 23. Year - S. mystery of overhead projector and fish-eye

You have maybe noticed, that in overhead projectors is often used very special lens, which looks more like a grooved plate. It is created in such way, that normal planoconvex lens is cut in concentric circles, the „end“ is kept and the result is again assembled. Finally we have axially symmetrical hilly glass (see figure).

Such lens has identical curvature to the original lens, and, according to Snell law we would expect, that will focus the light in the same way as original lens. However, looking at the situation using Fermat principle, the different beam paths do not experience the same time, as we have removed in different places different glass thickness. For example the shortest time is represented by the light beam travelling along the optical axis. It seems, that Fermat principle is failing, according it the lens should focus only the light following the optical axis and will not function as it should. Decide, who is correct: Snell or Fermat? And why?

Find the path of beams in two-dimensional situation, when the dependence of refractive index is described by a function ($r$ is distance from the origin):

$n(r)=n_{0}⁄(1+(r⁄a))$.

- (
**Bonus:**If the point source of light is placed into the environment with varying refractive index, then some light can be focused into a single point, similar as in the situation of converging lens. This point is then called image of original point source. Describe the geometrical transformation from source to the image, which is induced by the environment from the previous question.

Z Kroniky Dalimilovy.

### 1. Series 23. Year - S. Petrin mirror maze

- What will you see when standing between two vertical mirrors connected at right angle?

Lets have plane mirror inclined at angle 45°, moving to the left at the speed $v$. From the right there is a light ray of speed $c$ (e.g. angle of incidence is 45°) and is reflected upwards. Using Huygens principle calculate angle between incoming and reflected ray, e.g. correct the law of reflection for mowing mirrors.

Z dílny Dalimilovy.

### 3. Series 22. Year - S. ccccceeeee

* Imagine a strong laser at wavelength 400 nm, and shine it at the Moon. On its surface the light will reflect and come back. Assuming circular orifice of diameter of 1 cm through which the beam is going, what will be the diameter of the reflection on the Earth?
**Hint:** It will be much more, than 1 cm.

- In this task assume, that the aether really exists and predict, what will happen, if Mr. Michelson would make its measurement by other means: one arm would be 5 meters long and other 10 meters long. Such apparatus would create some interference pattern. Then he would rotate whole experiment by 90$°$, so both arms changed its positions. During rotating the experiment, we would see changes in interference patterns (assume rotating doubleslit). How would the interference patterns move at above rotation? How long would have to be the longer arm to inverse the interference fringes (e.g. minima would become maxima)?

- In the following task again assume existence of aether and that a body moving in aether is pulling it completely with to body, so the relative speed of aether to the body is zero. What would be then phase shift between two beams in the system in above figure? The light is splitted at semitransparent mirror into two beams and continues at perfectly rectangular path back to the semitransparent mirror, where it reaches screen at which interference fringes are observed. On the way are both beams three times reflected by a mirror and are going through the cylinder of length $L$, filled with water. Whole system is moving relatively to aether at speed $v$ to the right (do not forgot, that the cylinder is not moving relatively to aether!).

Zadali autři seriálu.

### 1. Series 20. Year - 4. captains diary

Contribute by some interesting record to the diary of the expedition (image, artistic creature, adventure story of length of daily observation, physical observation, …).

Napadlo Honzu Prachaře.

### 3. Series 15. Year - 4. accuracy of GPS

The Global Positioning System (GPS) is based on a rather simple principle. Satellites on 12-hour orbits emit perfectly synchronized signals then detected by a receiver. The receiver cannot carry perfect clock and therefore can detect only differences of distances to the satellites (i.e. it cannot measure the distance, but only the difference between two distances). Four satellites are enough to calculate the position.

Explain, why is the accuracy of GPS significantly better in the horizontal direction than in the vertical direction.

Při hledání informací o GPS zaujalo Honzu Houšťka.