An imaginary particle has a charge equal to that of an electron and mass 100 times the 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 expression for the radius of `n^(th)` 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_(rho)V^(2))/(r_(n))=(1)/(4 pi epsilon_(0))(Ze^(2))/(r_(n)^(2))`.....(i)
The quantization of angular momentum gives,
`m_(rho)Vr_(n)=(nh)/(2pi)`……(ii)
Solving Eqs. (i) and (ii), we get
`r=(n^(2)h^(2)epsilon_(0))/(z pi m_(p)e^(2)`
Substituting `m_(p)=100m`
where `m=` mass of electron and `z=4`
we get, `r_(n)=(n^(2)h^(2)epsilon_(0))/(400 pi me^(2))`
(b) As we know,
Energy of hydrogen atom in ground state `= 13.60 eV`
and `E_(n)prop(Z^(2)/(n^(2)))m`
For the given paritcle, `E_(4)=((-13.60)(4)^(2))/((4)^(2))xx100= -1360 eV`
and `E_(2)=((-13.60)(4)^(2))/((2)^(2))xx100 = -5440 eV`
`DE=E_(4)-E_(2)=4080 eV`
`lambda ("in"Å)=(12400)/(4080)=3.0Å`