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Question

An electron, an α particle and a proton have the same kinetic energy. Which of these particles has the shortest de Broglie wavelength?

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Solution

Kinetic energy is given by E=12mv2
Same kinetic energy implies that electron has highest velocity (as mass is lowest).
Assume mass of electron as 1 unit. Now, mass of proton approx 1836 and mass of alpha particle is 4 times that is 7344.

Now calculate velocity in each case
For electron v=2E, proton v=2E1836 and alpha particle v=2E7344

Also de-Broglie wavelength is given by λ=hmv

Now calculate momentum in each case
For electron mv=2E×1 proton mv=2E1836×1836 and alpha particle mv=2E7344×7344

Clearly electron has least momentum. Hence it has largest de-Broglie wavelength. And Alpha particle has least wavelength

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