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Dual Nature of Radiation question

2024 · 1 Feb · Shift 1 · Q72

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Dual Nature of Radiation
2024 · 1 Feb · Shift 1 · Q72

Dual Nature of Radiation question

2024 · 1 Feb · Shift 1 · Q72

JEE MainPhysicsDual Nature of RadiationMCQ+4 / −1

The de Broglie wavelengths of a proton and an

\alpha

particle are

\lambda

and

2 \lambda

respectively. The ratio of the velocities of proton and

\alpha

particle will be :

A
$8: 1$
B
$1: 2$
C
$1: 8$
D
$4: 1$

View written solutionFree

Correct answer: A

Use de Broglie relation

For a particle,

\lambda = \frac{h}{p} = \frac{h}{mv}

for non-relativistic motion.

So,

v = \frac{h}{m\lambda}

Thus velocity is inversely proportional to $m\lambda$ :

v \propto \frac{1}{m\lambda}

Write data for proton and $\alpha$ particle

Proton wavelength: $\lambda_p = \lambda$
Alpha particle wavelength: $\lambda_\alpha = 2\lambda$

Also,

m_\alpha = 4m_p

Find ratio of velocities

Using

v = \frac{h}{m\lambda}

we get

\frac{v_p}{v_\alpha} = \frac{h/(m_p\lambda)}{h/(m_\alpha\cdot 2\lambda)}

Cancel $h$ and $\lambda$ :

\frac{v_p}{v_\alpha} = \frac{m_\alpha \cdot 2}{m_p}

Since $m_\alpha = 4m_p$ ,

\frac{v_p}{v_\alpha} = \frac{4m_p\cdot 2}{m_p} = 8

Therefore,

v_p : v_\alpha = 8:1

Check options

A: $8:1$ ✅
B: $1:2$ ❌
C: $1:8$ ❌
D: $4:1$ ❌

So the correct option is A.

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