In a mass spectrometer used for measuring the masses of ions, the ions are initially accelerated 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 \(\left(\frac{\text { charge on the ion }}{\text { mass of the ion }}\right)\) will be proportional to

The radius of the orbit in which ions moving is determined by the relation as given below
\(
\frac{m v^{2}}{R}=q v B
\)
where \(m\) is the mass, \(v\) is velocity, \(q\) is charge of ion and \(B\) is the flux density of the magnetic field, so that \(q v B\) is the magnetic force acting on the ion, and \(\frac{m v^{2}}{R}\) 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
\(\omega=\frac{v}{R}=\frac{q B}{m}=2 \pi v\)
where \(v\) is frequency. Energy of ion is given by
or
\(
\begin{aligned}
E &=\frac{1}{2} m v^{2}=\frac{1}{2} m(R \omega)^{2} \\
&=\frac{1}{2} m R^{2} B^{2} \frac{q^{2}}{m^{2}} \\
E &=\frac{1}{2} \frac{R^{2} B^{2} q^{2}}{m}
\end{aligned}
\) ... (i)
If ions are accelerated by electric potential \(V\), then energy attained by ions
\(
E=q V ...(ii)
\)
From Eqs. (i) and (ii), we get
\(
q V=\frac{1}{2} \frac{R^{2} B^{2} q^{2}}{m}
\)
or
\(
\frac{q}{m}=\frac{2 V}{R^{2} B^{2}}
\)
If \(V\) and \(B\) are kept constant, then
\(
\frac{q}{m} \propto \frac{1}{R^{2}}
\)