Search

• blood sedimentation

Given a test tube full of human blood, how long does it take for the red cells to settle at the bottom (the so called blood sedimentation)? The usual method for such a measurement is to let the blood cells settle for an hour and than to measure the height of the cells that are already at the bottom (usually about 10 mm). We are interested in an approximate calculation. You may need to use the Stoke's relation $F=6$ π η r v$ and the value of dynamical viscosity of blood plasma $η$ = 2 Ns/m²$$.

• different eyes

Aleš was sitting in a tram and the Sun was about 60° to the left. Because he was staring at a hot blond in front of him one of his eyes was in the shadow of his nose. When the blond noticed he was staring at her he turned his eyes to the right and he found out that he saw different color shades in each of his eyes. Describe the difference between the shades he observed with his left eye compared to his right eye. Why did this happen?

To display a cyan blue on your monitor it has to light up both blue and red segment. If you however mix blue and red temperas you see that the resulting color is purple. Imagine that the temperas consist of small pieces and describe how does the color of a mixture of blue and red temperas depends on the size of these pieces.

Experimentally determine the dependance of transparency of a paper on the incidence angle of light.

Most of you have probably heard about the Young's double slit experiment. Have you, however, ever tried to reproduce this experiment and see the interference patterns for yourselves? There are also mechanical analogies to this experiment. For example you can observe the interference of two waves in water or two sound waves. Choose one or more of these experiments and measure the interference pattern. Then you can calculate the wave length and the speed of wave propagation. Photos of your apparatus will be welcomed!

• The index of refraction in a nonlinear medium varies with the intensity of light $I$ as $n=n_{1}+n_{2}I$, where $n_{1}$ and $n_{2}$ are positive constants. Describe the behaviour of a ray of light of given width passing through this medium, assuming the light intensity decreases as we go from the centre to the edges of the ray. (Qualitative description is acceptable, an analytic model and solution will obtain extra credit.)

• A slab of width $a$ consists of 2$N$ parallel neigbouring slabs (with no gaps) with alternating indices of refraction $n_{1}$ and $n_{1}$. A light wave is incident perpendicularly on the front slab. What is the effective index of refraction of this composite slab for $N→∞?$ Can you think of a physical reason why?

Hint: for any real matrix $A$ <p style=„text-align:center;“> lim_{$N→∞}(I+A/N)^{N}=\exp(A)$,

where $I$ is the identity matrix and exp($A)=I+A+A^{2}/2!+A^{3}/3!+\ldots]$.

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

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.

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.

• 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

• 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.