Question

Find the frequency of revolution of the electron in the classical model of the hydrogen atom. In what region of the spectrum are electromagnetic waves of this frequency?

Answer 1

$\nu=\frac{v_n}{2\pi r_n}$

$mv_nr_n=\frac{nh}{2\pi}\to v_n=\frac{nh}{2\pi mr_n}$

$m\frac{v_n^2}{r_n}=\frac{q^2}{4\pi\epsilon_0r_n^2}\to r_n=\frac{h^2\epsilon_0}{\pi m q^2}\cdot n^2$

$v_n=\frac{nh}{2\pi mr_n}=\frac{nh}{2\pi m}\frac{\pi m q^2}{h^2\epsilon_0n^2}=\frac{q^2}{2nh\epsilon_0}$

$\nu_n=\frac{v_n}{2\pi r_n}=\frac{q^2}{2\pi \cdot2nh\epsilon_0}\cdot \frac{\pi m q^2}{h^2\epsilon_0n^2}=$

$=\frac{q^4 m}{4h^3\epsilon_0^2}\cdot\frac{1}{n^3}$

For $n=1$ $\nu_1=\frac{(1.6\cdot10^{-19})^4\cdot9.1\cdot10^{-31}}{4\cdot (6.62\cdot10^{-34})^3\cdot(8.85\cdot10^{-12})^2}\cdot\frac{1}{1^3}\approx6.56\cdot10^{15}\ (Hz)$

which corresponds to ultraviolet region.