In mass spectrometer used for measuring the masses of ions, the ions are initaily accerlerated by an electric potential `V` and then made to describe semicircular paths of radius `R` using a magnetic field `B`.if `V` and `B` are kept constant, the ratio `(("charg e on the ion")/("mass of the ion"))` will be propertional to:

Options

A. `1//R`
B. R
C. `R^(2)`
D. `1//R^(2)`

Answer

D

Explanation

The radius of the orbit in which ions moving `R = (mv)/(qB)`
The angluar frequency of rotation of the ions about the vertical field `B` is given by
`omega = (v)/(R) = (qB)/(m) = 2pi v`
where `v` is frerquency.
Energy of ion is given by
`E = (1)/(2) mv^(2) = (1)/(2) m(R omega)^(2) = (1)/(2) mR^(2) B^(2) (q^(2))/(m^(2))`
or `E = (1)/(2) (R^(2) B^(2) q^(2))/(m)` .....(i)
If ions are accelerated by electric potental `V`, then energy attained by ions
`E = qV` ....(ii)
From Eqs. (i) and (ii) and get
`qV = (1)/(2) (R^(2) B^(2) q^(2))/(m)`
or `(q)/(m) = (2V)/(R^(2) B^(2))`
If `V` and `B` are kept constant, then `(q)/(m) prop (1)/(R^(2))`