# (a)  Using Bohr's postulates, derive the expression for the orbital     period of the electron moving in the $n^{th}$ orbit of hydrogen atom. (b)  Write Rydberg's formula for wavelengths of the spectral lines of hydrogen atom spectrum, Mention to which series in the emission spectrum of hydrogen ,$H_{\alpha }$ line belongs.

(a) The orbital period of an electron in a Hydrogen atom is given as;
$T= \frac{2\pi r_{n}}{v_{n}}-\left ( i \right )$
from Bohr's postulates, we have
$mv_{n}r_{n}= \frac{nh}{2\pi }$
where, $r_{n}= radius \: of\: n^{th}\, orbit$
$V_{n}= velocity \, of\, n^{th}\, orbit$
$m= mass.$
we have,

$r_{n}= \frac{n^{2}h^{2}\varepsilon _{0}}{\pi m_{e^{2}}}$
and velocity of the electron in $n^{th}$ orbital
$V_{n}= \frac{e^{2}}{2\varepsilon _{0}nh}$
where,h is plank's constant
on substituting the values of $r_{n}$ and $\frac{1}{n}$  in equation i we get,
$T= \frac{\frac{2\pi n^{2}h^{2}\varepsilon _{0}}{\pi me^{2}}}{\frac{e^{2}}{2\varepsilon _{0}nh}}$
$T= \frac{4n^{2}h^{2}\varepsilon _{0}}{me^{4}}$
(b) The Rydberg's formula for wavelengths of the spectral lines is given as;
$\frac{1}{\lambda }= R\left ( \frac{1}{n_{1}^{2}} -\frac{1}{n_{2}^{2}}\right )$ where,R is Rydberg's constant and $H_{\alpha }$ linebelongs to Balmer Series.

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