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?

1. λ = 10 nm
2. λ = 10^–1 nm
3. λ = 10^–4 nm
4. λ = 10^–6 nm

पर्याय

• a and c

• a and d

• c and d

• a and b

Answer 1

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.