In a mass spectrometer used for measuring the masses of ions, the ions are initally accelearted by an electric potential `V` and then mole to describe semicircular paths of radius `R` using a magnetic field `B`. If `V` and `B` are kept cosntant, the ratio `(("cha r g e on the iron")/("mass of the ion"))` will be porportional to

Options

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

Answer

B

Explanation

(b) Centripetal force is provided by the magnetic force `qvB`.
The radius of the orbit in which ions moving is determined by the relation as given.
`(mV^(2))/(R ) = qvB`
Where `m` is the mass, `v`is velocity, `q` is charge of ion and `B` is the flux density of the magentic field, so that `qvB` is the magnetic force acting on the ion, and `(mv^(2))/(R )`
is the centripetal force on the ion moving in a curved path of radius `R`.
The angular frequency of rotation of the ions about the vertical field `B` is given by
`Omega = (v)/(R ) = (qB)/(m) = 2 pi v`
Where `v` is frequency.
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 potential `V`, then energy attained by ions
`E = q V`
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))`