In a cyclotron, for the same values of B and R, what is the ratio of the maximum kinetic energies of α particles and deuteron? {Mass of deuteron = 2 m p , charge on deuteron = + e }

Question

In a cyclotron, for the same values of B and R, what is the ratio of the maximum kinetic energies of α particles and deuteron? {Mass of deuteron = 2 m p ​ , charge on deuteron = + e }

TestHub · Accepted Answer

The maximum kinetic energy of a particle in a cyclotron is given by $K = \frac{q ^{2} B ^{2} R ^{2}}{2 m}$ .

For an $α$ particle, charge $q_{α} = 2 e$ and mass $m_{α} = 4 m_{p}$ .

$K_{α} = \frac{( 2 e ) ^{2} B ^{2} R ^{2}}{2 ( 4 m _{p} )} = \frac{4 e ^{2} B ^{2} R ^{2}}{8 m _{p}} = \frac{e ^{2} B ^{2} R ^{2}}{2 m _{p}}$

For a deuteron, charge $q_{d} = e$ and mass $m_{d} = 2 m_{p}$ .

$K_{d} = \frac{e ^{2} B ^{2} R ^{2}}{2 ( 2 m _{p} )} = \frac{e ^{2} B ^{2} R ^{2}}{4 m _{p}}$

The ratio of their maximum kinetic energies is:

$\frac{K _{α}}{K _{d}} = \frac{\frac{e ^{2} B ^{2} R ^{2}}{2 m _{p}}}{\frac{e ^{2} B ^{2} R ^{2}}{4 m _{p}}} = \frac{4}{2} = 2$

Answer: $2$

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