An electron of charge e and mass m moving with an initial velocity `v_(0)hat(i)` is subjected to all electric field `E_(0)hat(j)`. The de-Broglie wavelength of the electron at a time t is
(Initial de-Broglie wavelength of the electron = `lambda_(0)`)

Options

A. `lambda_(0)`
B. `lambda_(0) sqrt(1 + (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))/(sqrt(1 + (e^(2) E_(0)^(2)t^(2))/(mv_(0)^(2))))`

Answer

C

Explanation

`therefore` de- Broglie relation of a charged particles,
` lambda = (h)/(mv)`
Velocity of charged particle at time t,
v = ` v_(0) hat(i) + (e E_(0))/(m ) t hat(j) or |v| = sqrt(v_(0)^(2) + ((eE_(0))/(m) t )^(2) ) `
Hence, `lambda = (h)/(m sqrt(v_(0)^(2) + ((eE_(0))/(m) t )^(2)) )`
or ` lambda = (h)/( mv_(0) sqrt( 1 + (e^(2) E_(0)^(2) t^(2))/(m^(2) v_(0)^(2)) )) or lambda = ( lambda_(0))/(sqrt( 1 + (e^(2) E_(0)^(2) t^(2))/(m^(2) v_(0)^(2)) )) `
`( because lambda_(0) = (h)/(mv_(0)))`