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

An electron (mass m)with an initial velocity `vecv=v_(0)hati` is in an electric field `vecE=E_(0)hatj` .
If `l,ambda_(0)=h..mv_(0)`.it's de-Broglie wavelength at time t is given by

A

`lambda_(0)`

B

`lambda_(0)sqrt((e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2)))`

C

`(lambda_(0))/(sqrt(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2))))`

D

`(lambda_(0))/(((1+e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2)))`

Text Solution

Verified by Experts

The correct Answer is:
C
Velocity of electron at the end of time t,
`vecv_(t)=vecv_(0)+vecat`
`therefore vecv_(t)=v_(o)hati+(vecF)/(m)t`
`therefore vecv_(t)=v_(0)hati-((eE_(0))/(m))t `
`therefore vecv_(t)=v_(0)hati-((eE_(0))/(m)t)`
`therefore =sqrt(v_(0)^(2)+(e^(2)E_(0)^(2))/(m^(2))t^(2))`
`=sqrt(V_(0)^(2)(1+(e^(2)E_(0)^(2))/(m^(2)v_(0)^(2))t^(2))`
`v_(t)=v_(0)sqrt(1+(e^(2)E_(0)^(2))/(m^(2)v_(0)^(2))t^(2)` .....(1)
de-Broglie wavelength,
`lambda=(h)/(mv)=lambdaprop(1)/(v)`
`therefore (lambda_(t))/(lambda_(0))=(v_(0))/(v_(t))`
`lambda_(t)=(lambda_(0)v_(0))/(v_(0)xxsqrt(1+(e^(2)E_(0)^(2))/(m^(2)v_(0)^(2)))t^(2))`
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