Problem Statement of Series 3, Year 39

Danka is interested in how the stability of a toilet-paper roll changes as part of the paper is used. She therefore took a new roll containing $N = 120$ sheets and measured its parameters. She found that the roll has height $h = 9.5\,\mathrm{cm}$, diameter $d_1 = 12.5\,\mathrm{cm}$, and mass $m_1 = 139\,\mathrm{g}$. The cardboard tube on which the paper is wound has a diameter $d_0 = 3.5\,\mathrm{cm}$ and mass $m_0 = 4\,\mathrm{g}$. By what factor does the potential energy required to tip the roll decrease when only one-third of the paper remains compared to the initial state? Assume the layers of paper are wound uniformly and tightly, without volumetric deformation.

We have a circuit with resistors arranged as shown in the diagram. The parallel configuration consists of infinitely many branches. In the $N$-th branch, there are $2^N$ resistors connected in series, each with a resistance of $R = 1\,\mathrm{\Omega}$. What is the total resistance between points A and B?

Lego was eating a banana. During the entire time he was eating, an unshielded gamma radiation source was located approximately 10 bananas away from him. This source irradiated Lego only while he was eating the banana, and was shielded at all other times. What is the activity of this source, given that it delivered the same radiation dose to Lego as the banana he ate? Estimate the properties of the bananas, the radiation source, and Lego, and express the result in SI units.

Consider an axially symmetric magnetic field between two large „magnetic mirrors“. The magnetic field lines connecting one mirror to the other become denser as they approach the edges. In all areas, the component of the magnetic field parallel to the axis of symmetry is much stronger than the perpendicular component, i.e., $B_\parallel \gg B_\perp$. At the midpoint between the mirrors, the magnetic induction reaches its minimum value $B_{\mathrm{min}}$, while near the mirrors it reaches its maximum $B_{\mathrm{max}}$.

For a charged particle moving in such a field, its magnetic moment $\mu = E_{\mathrm{kin,k}}/B$ is conserved, where $E_{\mathrm{kin,k}}$ is the kinetic energy associated with motion perpendicular to the symmetry axis. A particle is fired from the center at an angle $\theta$ with respect to the axis of symmetry. What is the condition for the particle to be reflected between the mirrors and remain trapped?

Suppose we fire a large number of particles from the center in random directions; that is, each direction on an imaginary sphere is equally probable. What fraction of these particles will remain confined between the mirrors?

A closed Petri dish with a radius of $2.0\,\mathrm{cm}$ containing $6.0\,\mathrm{ml}$ of ethanol is placed into a high-vacuum chamber with a volume of $30\,\mathrm{l}$. The dish is then opened, exposing the surface of the ethanol. How much will the ethanol level drop after a long period, and how will the height of the ethanol change over time? The temperature of the chamber and ethanol is $20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$.

This problem has an open solution, so be sure to cite all sources used.

Consider a short burst of electromagnetic radiation. What would be its minimum energy required to destroy the life on Earth?

Measure the cooling rates of three identical cups, each containing the same volume of different liquids over time. One cup should contain pure water, while the other two contain water with varying starch concentrations. Ensure the starch is well dissolved at a high temperature. The temperature should be measured at the bottom of the liquid.

1. Determine the plasma parameters of a flame and decide whether it is a plasma. – 3 points

2. In the attached file¹, you will find 4 measurements made with a Hairpin probe. The first measurement is performed in a vacuum, the others in plasma. Determine the probe length $L$, the resonance frequency $f_r$, and the $Q$ factor of the individual resonance peaks. Determine the electron concentration in the plasma and the plasma frequency for the individual measurements from the shift of the resonance frequency. – 3 points

3. In the attached file², you will find 5 measurements performed using a Curling probe with a spiral length of $L=17.5\,\mathrm{mm}$. Use the first four measurements to calibrate the probe, and for the fifth measurement, determine the electron concentration in the plasma and its relative permittivity. – 4 points