Никола 2019, 5 май — Задача 1. Problem 3. Bohr’s model of a hydrogen atom

Задачи – 5 Май 2019г. · 5 май 2019 г. · 10 т.

Условие

Consider a hydrogen atom, which consists of a proton, whose mass can be assumed virtually infinite, and an electron of mass $m_e = 9.11\times 10^{-31}$ kg. The proton has a positive charge of $e = 1.61\times 10^{-19}$ Cl and the electron has a negative charge $-e$, so that the atom is totally neutral.

1. It is known that most of the atoms are stable and can exist for a long time. This means that the electron being in the hydrogen atom cannot be at rest since the attraction force would inevitably make it fall onto the proton. Let the distance between the electron and the proton be equal to $r_0 = 5.00\times 10^{-11}$ m/ Assuming that the laws of classical physics are applicable, find the time $t_1$ that the initially motionless electron needs to fall onto the proton. [2 т.]

Hence, to guarantee the stability of the hydrogen atom, the electron must move around the proton like the planets revolve around the Sun. Let the electron orbit be circular and assume again that the laws of classical physics can still be applied.

2. Find the electron velocity $\mathrm{v}$ as a function of the orbit radius $r$. [1 т.]

3. Find the angular momentum $L$ of the electron as a function of the orbit radius $r$. [0,5 т.]

Danish physicist Niels Bohr postulated that the angular momentum of the electron is an integer of the Planck constant $\hbar = 1.05\times 10^{-34}$ Дж·с, that is $L = n\hbar$, where $n$ is the main quantum number.

4. Find the possible radii $r_n$ of the electron orbits in the hydrogen atom. [0,5 т.]

5. Calculate the numerical value of the minimum radius $r_n$ of the electron orbit in the hydrogen atom. [0,5 т.]

6. Find the possible values of the total energy $E_n$ of the electron in the hydrogen atom. [1 т.]

7. Calculate the numerical value of the minimum total energy $E_1$ of the electron in the hydrogen atom. [0,5 т.]

According to classical electrodynamics, if the electron is accelerated, it loses its energy due to electromagnetic radiation. The power of the radiation is given by

where $a$ is the electron acceleration, $\varepsilon_0 = 8.85\times 10^{-12}$ F/m is the dielectric constant, $c = 3.00\times 10^{8}$ m / s is the speed of light in vacuum.

Electromagnetic radiation would lead to the fact that in the Bohr model the electron would fall onto the proton. In the following assume that the electron orbit is almost circular.

8. Assuming that at the time moment $t = 0$ the electron moves in the orbit with the radius $r_1$, find the dependence of the orbit radius on time $t$. [1,5 т.]

9. Find and calculate the time $\tau_1$ that is needed for the electron to fall from the orbit with the radius $r_1$ onto the proton. [0,5 т.]

10. How many revolutions around the proton can the electron make during its fall for the time interval $\tau_1$? [2 т.]

Решение

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Оригинал в Архива: astro-552019a.pdf