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An electron (mass m) with an initial vel...

An electron (mass m) with an initial velocity `v=v_(0)hat(i)(v_(0)gt0)` is in an electric field `E=-E_(0)hat(l)(E_(0)="constant"gt0)`. Its de-Broglie wavelength at time t is given by

A

`(lamda_(0))/((1+(eE_(0)t)/(mv_(0))))`

B

`lamda_(0)(1+(eE_(0)t)/(mv_(0)))`

C

`lamda_(0)`

D

`lamda_(0)t`

Text Solution

Verified by Experts

The correct Answer is:
A
(a) : Here, `vec(E)=-E_(0)hat(i),"initial velocity"vec(v)=v_(0)hat(i)`
Force action on electron due to electric field
`vec(F)=(-e)(-E_(0)hat(i))=eE_(0)hat(i)`
Acceleration produced in the electron, `vec(a)=(vec(F))/(m)=(eE_(0))/(m)hat(i)`
Now, velocity of electron after time t,
`vec(v_(t))=vec(v)+vec(a)t=(v_(0)+(eE_(0)t)/(m))hat(i)or|vec(v_(t))|=v_(0)+(eE_(0)t)/(m)`
Now, `lamda_(t)=(h)/(mv_(t))=(h)/(m(v_(0)+(eE_(0)t)/(m)))=(h)/(mv_(0)(1+(eE_(0)t)/(mv_(0))))`
`=(+)/((1+(eE_(0)t)/(mv_(0))))(becauselamda_(0)=(h)/(mv_(0)))`
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