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Dual Nature of Radiation and Matter question

2018 · Q133
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Dual Nature of Radiation and Matter question

2018 · Q133

NEETPhysicsDual Nature of Radiation and MatterMCQ+4 / −1
An electron of mass m with an initial velocity v→=v0i^\overrightarrow v = {v_0}\widehat iv=v0​i (v0 > 0) enters an electric field E→=−E→0i^\overrightarrow E = - {\overrightarrow E _0}\widehat iE=−E0​i (E0 = constant > 0) at t = 0. If λ\lambdaλ0 is its de-Broglie wavelength initially, then its de- Broglie wavelength at time t is
  1. A
    λ0(1+eE0mv0t){{{\lambda _0}} \over {\left( {1 + {{e{E_0}} \over {m{v_0}}}t} \right)}}(1+mv0​eE0​​t)λ0​​
  2. B
    λ0(1+eE0mv0t){{\lambda _0}\left( {1 + {{e{E_0}} \over {m{v_0}}}t} \right)}λ0​(1+mv0​eE0​​t)
  3. C
    λ\lambdaλ0t
  4. D
    λ\lambdaλ0
View written solutionFree

Correct answer: A

Given, Initial velocity v = v0 i^\widehat ii

Electric field E = – E0i^\widehat ii

∴\therefore∴Initial de-Brogile wavelength,

λ\lambda λ0 = hmv0{h \over {m{v_0}}}mv0​h​

Now force due to electric field on electrons,

F→=(−e)(−E0i^)=eE0i^\overrightarrow F = \left( { - e} \right)\left( { - {E_0}\widehat i} \right) = e{E_0}\widehat iF=(−e)(−E0​i)=eE0​i

Acceleration produced in the electron,

a→=F→m=eE0mi^\overrightarrow a = {{\overrightarrow F } \over m} = {{e{E_0}} \over m}\widehat ia=mF​=meE0​​i

Now, velocity of electron after time t,

vt→=v→+a→t\overrightarrow {{v_t}} = \overrightarrow v + \overrightarrow a tvt​​=v+at = (v0+eE0tm)i^\left( {{v_0} + {{e{E_0}t} \over m}} \right)\widehat i(v0​+meE0​t​)i

⇒\Rightarrow⇒ ∣vt→∣=v0+eE0tm\left| {\overrightarrow {{v_t}} } \right| = {v_0} + {{e{E_0}t} \over m}​vt​​​=v0​+meE0​t​

Now, λ\lambda λt = hmvt{h \over {m{v_t}}}mvt​h​

= hm(v0+eE0tm){h \over {m\left( {{v_0} + {{e{E_0}t} \over m}} \right)}}m(v0​+meE0​t​)h​

= hmv0(1+eE0tmv0){h \over {m{v_0}\left( {1 + {{e{E_0}t} \over {m{v_0}}}} \right)}}mv0​(1+mv0​eE0​t​)h​

= λ0(1+eE0tmv0){{{\lambda _0}} \over {\left( {1 + {{e{E_0}t} \over {m{v_0}}}} \right)}}(1+mv0​eE0​t​)λ0​​ [As λ0=hmv0{\lambda _0} = {h \over {m{v_0}}}λ0​=mv0​h​]

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