b
The centripetal force is provided by the magnetic force.

i.e., \(\frac{\mathrm{mv}^{2}}{\mathrm{R}}=\mathrm{qvB}.........(1)\)

where \(m=\) mass of the ion, \(v=\) velocity, \(q=\) charge of ion, \(\mathrm{B}=\) flux density of the magnetic field.

we have, \(\mathrm{v}=\mathrm{R} \omega\)

or \(\omega=\frac{\mathrm{v}}{\mathrm{R}}=\frac{\mathrm{qB}}{\mathrm{m}} \quad(\mathrm{From}(1))\)

Energy of ion is given by,

\(\mathrm{E}=\frac{1}{2} \mathrm{mv}^{2}=\frac{1}{2} \mathrm{m}(\mathrm{R} \omega)^{2}=\frac{1}{2} \mathrm{mR}^{2} \frac{\mathrm{q}^{2} \mathrm{B}^{2}}{\mathrm{m}^{2}}\)

or \(\mathrm{E}=\frac{1}{2} \frac{\mathrm{R}^{2} \mathrm{B}^{2} \mathrm{q}^{2}}{\mathrm{m}}.........(2)\)

If ions are accelerated by electric potential \(V\), the energy attained by ions,

\(E=q V.........(3)\)

From eqns \(( 2)\) and \(( 3)\)

\(\mathrm{qV}=\frac{1}{2} \frac{\mathrm{R}^{2} \mathrm{B}^{2} \mathrm{q}^{2}}{\mathrm{m}}\) or \(\left(\frac{\mathrm{q}}{\mathrm{m}}\right)=\frac{2 \mathrm{V}}{\mathrm{R}^{2} \mathrm{B}^{2}}\)

i.e., \(\left(\frac{\mathrm{q}}{\mathrm{m}}\right) \propto \frac{1}{\mathrm{R}^{2}}(\text { If } \mathrm{V} \text { and } \mathrm{B} \text { are const. })\)