An electron of mass 'm' with an initial velocity enters an electric field . If the initial de-Broglie wavelength is , then its value after time t would be [2025]

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Q.
An electron of mass 'm' with an initial velocity $\vec{v} = v_{0} \hat{i} (v_{0} > 0)$ enters an electric field $\vec{E} = - E_{0} \hat{k}$ . If the initial de-Broglie wavelength is $λ_{0}$ , then its value after time t would be [2025]

1 $\frac{λ_{0}}{\sqrt{1 + \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$

2 $\frac{λ_{0}}{\sqrt{1 - \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$

3 $λ_{0}$

4 $λ_{0} \sqrt{1 + \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}$

Ans.
(1)

$\vec{v} = v_{0} \hat{i} - \frac{E_{0} e}{m} t \hat{k}$

$| \vec{v} | = \sqrt{v_{0}^{2} + \frac{E_{0}^{2} e^{2} t^{2}}{m^{2}}}$

$λ_{0} = \frac{h}{m v_{0}}$

$λ' = \frac{h}{m v_{0} \sqrt{1 + \frac{E_{0}^{2} e^{2} t^{2}}{v_{0}^{2} m^{2}}}}$

$λ' = \frac{λ_{0}}{\sqrt{1 + \frac{E_{0}^{2} e^{2} t^{2}}{v_{0}^{2} m^{2}}}}$

Solution

Q.
An electron of mass 'm' with an initial velocity $\vec{v} = v_{0} \hat{i} (v_{0} > 0)$ enters an electric field $\vec{E} = - E_{0} \hat{k}$ . If the initial de-Broglie wavelength is $λ_{0}$ , then its value after time t would be [2025]

1 $\frac{λ_{0}}{\sqrt{1 + \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$

2 $\frac{λ_{0}}{\sqrt{1 - \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$

3 $λ_{0}$

4 $λ_{0} \sqrt{1 + \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}$

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Solution

Q. An electron of mass 'm' with an initial velocity v→=v0i^(v0>0) enters an electric field E→=–E0k^. If the initial de-Broglie wavelength is λ0, then its value after time t would be [2025]

1 λ01+e2E02t2m2v02

2 λ01–e2E02t2m2v02

3 λ0

4 λ01+e2E02t2m2v02

Ans. (1) v→=v0i^–E0emtk^ |v→|=v02+E02e2t2m2 λ0=hmv0 λ'=hmv01+E02e2t2v02m2 λ'=λ01+E02e2t2v02m2

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Q.
An electron of mass 'm' with an initial velocity $\vec{v} = v_{0} \hat{i} (v_{0} > 0)$ enters an electric field $\vec{E} = - E_{0} \hat{k}$ . If the initial de-Broglie wavelength is $λ_{0}$ , then its value after time t would be [2025]

1 $\frac{λ_{0}}{\sqrt{1 + \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$

2 $\frac{λ_{0}}{\sqrt{1 - \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$

3 $λ_{0}$

4 $λ_{0} \sqrt{1 + \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}$