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

Answer

B

Explanation

(b) If an ion of charge `q` mass `m` moves in magnetic fied of flux density `B` with velocity `v`, its path is circular. The magnetic force `qvB` acting on the ion provides required centripeteal force `(mv^(2))/(R)` on the ion moving in a circular path of radius `R`.
`(mv^(2))/(R) = qvB implies R = (mv)/(qB)`
The angular frequency of rotation of the ions about the vertical field `B` is given by `omega = (v)/(R) = (qB)/(m)`
Energy of ion is given by `E = (1)/(2) mv^(2) = (1)/(2) m(R omega)^(2)`
`implies E = (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 accelearated by electric potential `V`, then energy attained by ions,
`E = qV`
From Eqs.(I) and (ii), we 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))`