Problem database

There are two parallel disks in a vessel with a gas whose pressure can be changed. One disk hangs on an elastic thread, the other rotates at a constant angular velocity. Twist angle of the first disk at pressure $P1$ is equal to $\varphi_1$. As the gas pressure increases, the angle of twisting of the thread first increases, and then, having reached the value $\varphi_2$, it ceases to depend on the gas pressure. Explain this effect. How does the thread twist angle depend on the gas pressure at $\varphi \ll \varphi_2$?

Estimate the mass of liquid air evaporated in an hour from a poorly pumped Dewar vessel, if the air pressure (at a temperature of $293$ K) remaining between the vessel walls is $0.133$ Pa. The surface of the vessel is $600$ $cm^2$ , the specific heat of vaporization of liquid air is $0.2$ MJ / kg, its temperature is $81$ K. The gap between the vessel walls is small compared to the free path of the molecules.

Solve problems 5.8.5 if the initial tangent of the angle of incidence is $\frac{1}{m}$, where $m$ is an integer.

Atoms having the same velocity $v$ in modulus simultaneously fly into the cylinder through a small hole located in the center of the bottom of the cylinder. The directions of atomic velocities are distributed uniformly inside the cone with a small angle $\Delta$ at the vertex. The velocity cone is aligned with the cylinder. Radius of the cylinder $R$, its height $H$. Estimate the time of uniform filling of the space inside the cylinder with atoms in the case of elastic reflection of atoms from its wall and in the case when, after the time $t \gg \frac{R}{v}$, $\frac{H}{v}$ after hitting the wall, the atom moves inside the cylinder at any angle to its wall at a speed $v$.

One electrode of a flat capacitor is an emitter of electrons, the other consists of parallel wires, the gaps between which are significantly smaller than the distance $d$ between the electrodes. Potential difference between the electrodes $V$. Determine the spread of electrons passing through the second electrode by the "transverse" energy, if the gap between the wires is $a$, and their thickness is $b$. Ignore the initial electron velocity.

Determine the potential difference on the capacitor plates if a ribbon beam of protons perpendicular to the plates and passing through two narrow parallel slits is focused at a distance $l$ from the second plate. The protons were accelerated by the potential difference $V_0$. Distance between capacitor plates $d$. The first lining is grounded, $l \gg d$.

Prove that uniformly charged grid filaments with square cells focus a parallel electron beam that has passed through the cell to a point if the thickness of the filaments is much smaller than the cell size and the beam falls perpendicular to the grid plane. What is the focal length of such a cell if the electric field away from the grid plane is uniform and on the right is $E_1$, on the left is $E_2$, and the electron energy is $eV$ ?

A container filled with air at room temperature and atmospheric pressure is irradiated with X-rays that ionize a small portion of the molecules. Negative ions are $O_2$ molecules that have "captured" an electron. The size of the container is $10 \times 10 \times 2$ cm; two walls of $10 \times 10$ cm are made of metal, and the rest are made of insulating material. A voltage of $1000$ V is applied between the conducting walls, causing a current of $1.5$ uA. Assuming the number of positive and negative singly charged ions is the same, estimate the fraction of ionized gas molecules. The free path of ions is $10^{-7}$ m.

When a capacitor with a charge of $q$ is discharged, a mass $m$ of rattlesnake gas is released through an electrolytic bath with acidified water. The mass of the substance released during electrolysis depends only on the passed charge. So, by discharging the capacitor through $k$ series-connected baths, we get the mass $km$ of rattlesnake gas. By burning this gas, we will get a lot of energy. For a sufficiently large $k$, this energy will exceed the original energy of the charged capacitor! Therefore, in some ways our reasoning is wrong. Find this error.

The total current density in electrolytes is the sum of the current density of positive ions and the current density of negative ions: $j = e_+n_+v_+ + e_- n_- v_-$, where $e \pm$, $v \pm$ and $n \pm$ are the charge, velocity of positive and negative ions and their number per unit volume. Why is the mass of the substance released at the cathode proportional to the total current, and not to the current of only positive ions?

a. The formula for the transformation of the $\overrightarrow{E}$ and $\overrightarrow{B}$ fields when they move at the speed $\overrightarrow{\beta} c$ has the following form:

$\overrightarrow{E}' = \overrightarrow{E} {\parallel} + \gamma(\overrightarrow{E} {\perp} - [\overrightarrow{\beta} \times \overrightarrow{B} ]), \overrightarrow{B}' = \overrightarrow{B} {\parallel} + \gamma(\overrightarrow{B} {\perp} + (\overrightarrow{\beta} \times \overrightarrow{E} ]), \gamma = \frac{1}{\sqrt{1 - \beta^2}}$,

where $\overrightarrow{E}'$ and $\overrightarrow{B}'$ are the electric and magnetic fields in the drift; $\overrightarrow{E} {\parallel}$, $\overrightarrow{E} {\perp}$ and $\overrightarrow{B} {\parallel}$, $\overrightarrow{B} {\perp}$ — components electric and magnetic fields fields, parallel services and perpendicular lines $- c \overrightarrow{\beta}$ in the initial system. The movement of the $\overrightarrow{E}'$ and $\overrightarrow{B}'$ fields at a speed of $- c \beta$ returns the previous state. Check it out.

b. Using the field transformation formulas given in point $a$, solve the following problems: 14.3.1–14.3.3, 14.3.5.

c. Using the field transformation formulas given in point $a$, solve problems 14.3.6 a, b, and 14.3.7.

d. Prove that for $\beta \to 1$, the fields $\overrightarrow{E}'$ and $B'$ are perpendicular.

a.".. . For a moving electron, the electric field $E$ is equivalent to an additional magnetic field $\overrightarrow{B} = [\overrightarrow{\beta} \times \overrightarrow{E} ]$" (G. Bethe, E. Solipter. Quantum mechanics with one and two electrons, Moscow: Fizmatgiz, 1960). Determine, using this statement, the force acting on the magnetic moment of an electron in a hydrogen atom, if the electron) It's moving by circular orbit.

A charged capacitor suspended on a thread, it would seem, cannot move translationally together with the thread and the suspension, if the angle $\alpha$ is not straight, since the magnetic force of interaction of two jointly moving charges creates a rotational moment. This rotational moment could be detected experimentally if we assume that the capacitor moves with the Earth at a speed of $\beta c$. Is it so?

At what minimum potential difference in a flat capacitor do electrons accelerated by the potential $U = 1$ MV, flying into the capacitor through a small hole in the lower plate at an angle $\alpha = 30^\circ$ to it, do not reach the upper plate?

A magnetic field in a television tube rotates electrons with energy $E = 2 \cdot 10^4$ eV by an angle $\alpha = 60^\circ$ . The deflecting coil creates a magnetic field on a section of the tube with a length of $l = 10$ cm. Determine the magnetic field induction. What error is made when calculating the induction, if we ignore the changes in the mass of the electron during its movement?

Complete the task 14.4.23 if the area occupied by the magnetic field is moving perpendicular to its boundary with the velocity $\beta_1 c$.