Atomic Physics

Bohr's model and hydrogen spectrum

Quantisation
$m v r = \frac{nh}{2 π}$
Radius, speed, energy
$r_{n} = 0.529 \frac{n ^{2}}{Z}$ Å $,$ $v_{n} = 2.18 \times 1 0^{6} \frac{Z}{n}$ m/s $,$ $E_{n} = - 13.6 \frac{Z ^{2}}{n ^{2}}$ eV
Spectral lines
$\frac{1}{λ} = R Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$
Lyman $n_{1} = 1$ (UV), Balmer $n_{1} = 2$ (visible), Paschen $n_{1} = 3$ (IR).
Number of lines
$\frac{n ( n - 1 )}{2}$ (from level $n$ down to the ground state)
$K = - E,$ $U = 2 E$

Cut-off wavelength
$λ_{m i n} = \frac{h c}{e V}$
Moseley's law
$ν = a (Z - b)$
Not in the current JEE Main syllabus; kept for older papers.