If the exact orbital velocity of an electron in the $3^{ ext{rd}}$ Bohr orbit of a hydrogen atom is

Question

If the exact orbital velocity of an electron in the $3^{	ext{rd}}$ Bohr orbit of a hydrogen atom is denoted as '$v$', what will be its precise orbital velocity in the $1^{	ext{st}}$ Bohr orbit?

Professor PC Thomas Classes · Answer

(d) : According to Bohr's theory, the orbital velocity of an electron in the $n^{	ext{th}}$ orbit of a hydrogen-like species is inversely proportional to the principal quantum number $n$ ($v_n \propto \frac{Z}{n}$). For hydrogen ($Z=1$), $v_n \propto 1/n$. Therefore, $v_3 = v_1 / 3$. Since we are given that $v_3 = v$, it rigorously follows that $v_1 = 3 \cdot v_3 = 3v$.