According to Bohr's first postulate the kinetic energy $\frac{1}{2} m v^2$ of the electron in C.G.S. system is ( $\mathrm{m}=$ mass of electron, v is its velocity, r is the radius of the stationary orbit around the nucleus with charge Ze )

In the spectrum of hydrogen atom the ratio of the longest wavelength in the Balmer series to the longest wavelength in the Lyman series is

The series limit for the frequency of Balmer series is ' $\mathrm{v}_{\mathrm{B}}$ ', then the series limit frequency of Paschen series ' $v_p$ ' is

Radioactive materials A and B have decay constants ' $9 \lambda$ ' and ' $\lambda$ ' respectively. Initially they have same number of nuclei. The ratio of number of nuclei of material ' $A$ ' to that of ' $B$ ' will be $\left(\frac{1}{\mathrm{e}}\right)$ after time ' $t$ '. So ' $t$ ' is equal to