A proton and an -particle, having kinetic energies and respectively, enter into a magnetic field at right angles

Question

A proton and an $α$ -particle, having kinetic energies $K_{p}$ and $K_{α}$ respectively, enter into a magnetic field at right angles. The ratio of the radii of trajectory of proton to that of $α$ -particle is 2:1. The ratio of $K_{p} : K_{α}$ is:

Smart Achievers · Accepted Answer

(4)

$F = \frac{m V^{2}}{r} and F = q V B$

$∴$ $\frac{m V^{2}}{r} = q V B \Rightarrow r = \frac{m V}{q B}$

$or, r = \frac{\sqrt{2 m K}}{q B}$ $(∵ p = m V = \sqrt{2 m K})$ $\Rightarrow \frac{r^{2} q^{2} B^{2}}{2 m} = K$

$k_{p} = \frac{r_{p}^{2} q_{p}^{2} B^{2}}{2 m_{p}} and k_{α} = \frac{r_{α}^{2} q_{α}^{2} B^{2}}{2 m_{α}}$

$∴$ $\frac{K_{p}}{K_{α}} = \frac{r_{p}^{2} q_{p}^{2} m_{α}}{r_{α}^{2} q_{α}^{2} m_{p}} = {(\frac{2}{1})}^{2} {(\frac{1}{2})}^{2} (\frac{4}{1})$ $or, \frac{K_{p}}{K_{α}} = 4 : 1$

Solution

Q.
A proton and an $α$ -particle, having kinetic energies $K_{p}$ and $K_{α}$ respectively, enter into a magnetic field at right angles. The ratio of the radii of trajectory of proton to that of $α$ -particle is 2:1. The ratio of $K_{p} : K_{α}$ is:

1 1:8

2 8:1

3 1:4

4 4:1

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