First slide

Wave nature of matter

Question

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 paths are equal to each other
assuming the field induction vector $\vec{B}$ is perpendicular to the velocity vectors of the $α$ -particle and the proton is

Easy

A

1
B

$\frac{1}{4}$
C

$\frac{1}{2}$
D

2

Solution

When a charged particle (charge q, mass m) enters perpendicularly in a magnetic field (B) than, radius of the path
$r = \frac{m v}{q B} \Rightarrow m v = q B r$
Also, de Broglie wavelength, $λ = \frac{h}{m v}$
$\Rightarrow λ = \frac{h}{q B r} \Rightarrow \frac{λ_{α}}{λ_{p}} = \frac{q_{p} r_{p}}{q_{α} r_{α}} = \frac{1}{2} \times \frac{1}{1} = \frac{1}{2}$

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