An imaginary particle has a charge equal to that of an electron and mass 100 times tha mass of the electron. It moves in a circular orbit around a nucleus of charge + 4 e. Take the mass of the nucleus to be infinite. Assuming that the Bhor model is applicable to this system. (a)Derive an experssion for the radius of nth Bhor orbit. (b) Find the wavelength of the radiation emitted when the particle jumps from fourth orbit to the second orbit.

Explanation

(a) we have `(m_q v^2)/(r_n) = (1)/(4pi epsilon_0) (Ze^2`)/(r_n^2`) …(i)
The quantization of angular momentum gives
`m_pur_n = (nh)/(2pi) …(ii)`
Solving Eqs. (i) and (ii), we get
`r = (n^2h^2epsilon_0)/(Zpim_pe^2)`
`Substituting `m_p = 100m`
where, m = mass of electorn and Z= 4 we get `r_n = (n^2 h^2 epsilon_0)/(400pi me^2)`
(b) As we know, `E_1^H = - 13.60 eV`
and `E_n prop ((Z^2)/(n^2)) m`
For the given particle,`E_4 = (-13.60)(4)^2 )/((4))^2 xx100`
and E_2 = (-13.60)(4)^2/((2))^2` xx100
`=- 5440eV`
`DeltaE=E_4-E_2`
`=4080 eV`
`:. lambda(in Å) = (12375)/(DeltaE(in eV))`
`=(12375)/(4080)`
`=3.0 Å`