An electron of mass is moving in an electric field , with an initial velocity . If , its de-Broglie wavelength at time is

Question

An electron of mass $m$ is moving in an electric field $\vec{E} = - 2 E_{0} \hat{i} (E_{0} = constant > 0)$ , with an initial velocity $\vec{V} = v_{0} \hat{i} (v_{0} = constant > 0)$ . If $λ_{0} = \frac{h}{4 m v_{0}}$ , its de-Broglie wavelength at time $t$ is _______. ( $e$ = charge of electron) [2026]

Smart Achievers · Accepted Answer

(3)

$V = u + a t$

$V = v_{0} + (\frac{2 e E_{0}}{m}) t$

$λ = \frac{h}{m V} = \frac{h}{m [v_{0} + \frac{2 e E_{0} t}{m}]}$

$Put λ_{0} = \frac{h}{4 m v_{0}} \Rightarrow h = 4 λ_{0} m v_{0}$

$λ = \frac{4 m λ_{0} v_{0}}{m [v_{0} + \frac{2 e E_{0}}{m} t]} =$ $\frac{4 λ_{0}}{1 + \frac{2 e E_{0}}{m v_{0}} t}$

Solution

1 $\frac{4 λ_{0}}{[1 - \frac{E_{0} e}{2 m} \frac{t}{v_{0}}]}$

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

3 $\frac{4 λ_{0}}{[1 + \frac{2 E_{0} e}{m} \frac{t}{v_{0}}]}$

4 $\frac{4 λ_{0}}{[1 - \frac{2 E_{0} e}{m} \frac{t}{v_{0}}]}$

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Solution

Q. An electron of mass m is moving in an electric field E→=-2E0i^(E0=constant>0), with an initial velocity V→=v0i^(v0=constant>0). If λ0=h4mv0, its de-Broglie wavelength at time t is _______. (e = charge of electron) [2026]

1 4λ0[1-E0e2mtv0]

2 4λ0[1+E0e2mtv0]

3 4λ0[1+2E0emtv0]

4 4λ0[1-2E0emtv0]

Ans. (3) V=u+at V=v0+(2eE0m)t λ=hmV=hm[v0+2eE0tm] Put λ0=h4mv0⇒h=4λ0mv0 λ=4mλ0v0m[v0+2eE0mt]=4λ01+2eE0mv0t

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1 $\frac{4 λ_{0}}{[1 - \frac{E_{0} e}{2 m} \frac{t}{v_{0}}]}$

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

3 $\frac{4 λ_{0}}{[1 + \frac{2 E_{0} e}{m} \frac{t}{v_{0}}]}$

4 $\frac{4 λ_{0}}{[1 - \frac{2 E_{0} e}{m} \frac{t}{v_{0}}]}$