JEE Main 2025 Jan 24 Shift 1, Physics Q17: Wave Nature of Matter

JEE Main2025Jan 24, Shift 1Physics

An electron of mass ' $m$ ' with an initial velocity $v = v_{0} \hat{i} (v_{0} > 0)$ enters an electric field $E = - E_{0} \hat{k}$ . If the initial de Broglie wavelength is $λ_{0}$ , the value after time $t$ would be :-

A
$\frac{λ _{0}}{1 + \frac{e ^{2} E _{0}^{2} t ^{2}}{m ^{2} v _{0}^{2}}}$
B
$\frac{λ _{0}}{1 - \frac{e ^{2} E _{0}^{2} t ^{2}}{m ^{2} v _{0}^{2}}}$
C
$λ_{0}$
D
$λ_{0} 1 + \frac{e ^{2} E _{0}^{2} t ^{2}}{m ^{2} v _{0}^{2}}$

Solution

Force on electron: $F = - e E = - e (- E_{0} \hat{k}) = + e E_{0} \hat{k}$ . The acceleration is $a = \frac{e E _{0}}{m} \hat{k}$ .

After time $t$ , velocity components:

$v_{x} = v_{0}, v_{z} = \frac{e E _{0}}{m} t$

Speed:

$∣ v (t) ∣ = v_{0}^{2} + \frac{e ^{2} E _{0}^{2} t ^{2}}{m ^{2}} = v_{0} 1 + \frac{e ^{2} E _{0}^{2} t ^{2}}{m ^{2} v _{0}^{2}}$

De Broglie wavelength is $λ = h / (m v)$ . Initially $λ_{0} = \frac{h}{m v _{0}}$ . At time $t$ :

$λ (t) = \frac{h}{m ∣ v ( t ) ∣} = \frac{λ _{0}}{1 + \frac{e ^{2} E _{0}^{2} t ^{2}}{m ^{2} v _{0}^{2}}}$

$∴$ Option (1).

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JEE Main 2025 Jan 24 Shift 1, Physics Q17: Wave Nature of Matter

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