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

Options

A. `lamda_(0)`
B. `lamda_(0)sqrt(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2)))`
C. `(lamda_(0))/(sqrt(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2))))`
D. `(lamda_(0))/(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2)))`

Explanation

Initial de-Broglie wavelength of electron,
`" "lamda_(0)=(h)/(mv_(0))`
Force on electron in electric field, `" "F=-eE=-eE_(0)hatj`
Acceleration of electron, `" "a =F/m=(eE_(0)hatj)/(m)`
It is acting along negative y-axis.
The initial velocity of electron along x-axis, `v_(xo)=v_(0)hati.` Initial velocity of electron along y-axis, `" "v_(y0)=0.`
Velocity of electron after time t along y-axis,
`" "v_(y)=0+(-(eE_(0)hatj)/(m))t=-(eE_(0))/(m)thatj`
Magnitude of velocity of electron after time t is
`" "v=sqrt(v_(x)^(2)+v_(y)^(2))=sqrt(v_(0)^(2)+((-eE_(0))/(m)t)^(2))`
`implies" "=v_(0)sqrt(1+(e^(2)E_(0)^(2)t^(2))/(m^(2)v_(0)^(2)))`
de-Broglie wavelength, `" "lamda=(h)/(mv)`
`implies" "(h)/(mv_(0)sqrt(1+e^(2)E_(0)^(2)t^(2)//(m^(2)v_(0)^(2))))`
`" "(lamda_(0))/(sqrt(1+e^(2)E_(0)^(2)t^(2)//m^(2)v_(0)^(2)))`