An electron of mass `m` with an initial velocity
`vec(v) = v_(0) hat`(i) `(v_(0) gt 0)` enters an electric field
`vec(E ) =- E_(0) hat (i)` `(E_(0) = constant gt 0)` at `t = 0` . If `lambda_(0)` is its de - Broglie wavelength initially, then its de - Broglie wavelength at time `t` is

Options

A. `lambda_(0)/((1+(eE_(0))/mt/v_(0)))`
B. `lambda_(0)(1+(eE_(0)t)/(mv_(0)))`
C. `lambda_(0)`
D. `lambda_(0)t`

Answer

A

Explanation

Initial de-Broglie wavelength of electron, `lambda_(0)=h/(mv_(0))`…(i)
Force on electron in electrid field, `F=-eE=-e[-E_(0)hati]=eE_(0)hati`
Acceleration of electron `a=F/m=(eE_(0)hati)/m`
Velocity of electron after time `t`, `v=v_(0)hati+((eE_(0)hati)/m)t=(v_(0)+(eE_(0))/mt)hati`
`=v_(0)(1+(eE_(0))/(mV_(0))t)hati`
de-Broglie wavelength associated with electron at time `t` is `lambda=h/(mv)=h/(m[v_(0)(1+(eE_(0))/(mv_(0))t)])=lambda_(0)/[[1+(eE_(0))/(mv_(0))t])`
`[therefore lambda_(0)=h/(mv_(0))]`