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Question
Relativistic corrections become necessary when the expression for the kinetic energy `1/2 mv^2`, becomes comparable with mc2, where m is the mass of the particle. At what de Broglie wavelength will relativistic corrections become important for an electron?
- λ = 10 nm
- λ = 10–1 nm
- λ = 10–4 nm
- λ = 10–6 nm
Options
a and c
a and d
c and d
a and b
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Solution
c and d
Explanation:
De-Broglie or matter wave is independent of die charge on the material particle. It means matter wave of the de-Broglie wave is associated with every moving particle (whether charged or uncharged).
The de-Broglie wavelength at which relativistic corrections become important is that the phase velocity of the matter waves can be greater than the speed of the light (3 × 108 m/s).
The wavelength of de-Broglie wave is given by λ = h/p = h/mv
Here, h = 6.6 × 10-34 Js
And for electron, m = 9 × 10-31 kg
To approach these types of problems we use the hit and trial method by picking up each option one by one.
In option (a), λ1 = 10 nm = 10 × 10–9 m = 10–8 m
⇒ `v_1 = (6.6 xx 10^-34)/((9 xx 10^-31) xx 10^-8)`
= `2.2/3 xx 10^5 = 10^5` m/s
In option (b), λ2 = 10–1 nm = 10–1 × 10–9 m = 10–10 m
⇒ `v_2 = (6.6 xx 10^-34)/((9 xx 10^-31) xx 10^-10) = 10^7` m/s
In option (c), λ3 = 10–4 nm = 10–4 × 10–9 m = 10–13 m
⇒ `v_3 = (6.6 xx 10^-34)/((9 xx 10^-31) xx 10^-13) = 10^10` m/s
In option (d), λ4 = 10–6 nm = 10–6 × 10–9 m = 10–15 m
⇒ `v_4 = (6.6 xx 10^-34)/((9 xx 10^-31) xx 10^-15) = 10^12` m/s
Thus options (c) and (d) are correct as v3 and v4 is greater than 3 × 108 m/s.
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