Radii of the Orbits

Bohr's first two postulates together are enough to work out the size of each allowed orbit. Let the electron (mass $m_e$) move with speed $v_n$ in a circular orbit of radius $r_n$, the nth stable orbit. The second postulate fixes its angular momentum as $m_ev_nr_n=\frac{nh}{2\pi}$.

The first postulate says the centripetal force keeping the electron in this circular orbit comes from the electrostatic attraction between the electron (charge $-e$) and the nucleus (charge $+Ze$, for an atom with atomic number Z): $\frac{m_ev_n^2}{r_n}=\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r_n^2}$.

These are two equations in the two unknowns $v_n$ and $r_n$. Eliminating $v_n$ between them gives the orbit radius directly: $r_n=\frac{n^2h^2\epsilon_0}{\pi m_eZe^2}$ -- showing immediately that the radius grows as the SQUARE of the principal quantum number n, so higher orbits are very much larger than lower ones, not just slightly larger. Eliminating $r_n$ instead gives the orbital speed, $v_n=\frac{Ze^2}{2\epsilon_0hn}$, which DECREASES as n increases -- electrons in higher, larger orbits move more slowly, not faster. …