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  4. The ratio of de-Broglie wavelength of a α -particle to that of a proton being subjected to the same magnetic field so that the radii of their path are equal to each other assuming the field induction vector vecB is perpendicular to the velocity vectors of the α -particle and the proton is

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Q. The ratio of de-Broglie wavelength of a $\alpha$ -particle to that of a proton being subjected to the same magnetic field so that the radii of their path are equal to each other assuming the field induction vector $\vec{B}$ is perpendicular to the velocity vectors of the $\alpha$ -particle and the proton is

Dual Nature of Radiation and Matter

Solution:

When a charged particle (charge q, mass $m$ ) enters perpendicularly in a magnetic field (B) than, radius of the path described by it
$r=\frac{m v}{q B} \Rightarrow m v=q B r$.
Also de-Broglie wavelength $\lambda=\frac{h}{m v}$
$\Rightarrow \lambda=\frac{h}{q B r}$
$\Rightarrow \frac{\lambda_{\alpha}}{\lambda_{p}}=\frac{q_{p} r_{p}}{q_{\alpha} r_{\alpha}}=\frac{1}{2}$