An electron (mass m) with an initial velocity $v=v_0\hat i(v_0>0)$ is in an electric field $E=-E_0\hat i$ ($E_0$ = constant > 0). Its de-Broglie wavelength at time t is given by:
Answer & explanation
Correct answer: option 1
Initial de-Broglie wavelength of electron, $λ_0=\frac{h}{mv_0}$ (i)
Force on electron in electric field, $F=-eE=-e[-E_0\hat i]=eE_0\hat i$
Acceleration of electron, $a=\frac{F}{m}=\frac{eE_0\hat i}{m}$
Velocity of electron after time t, $v=v_0\hat i+(\frac{eE_0\hat i}{m})t=(v_0+\frac{eE_0}{m}t)\hat i=v_0(1+\frac{eE_0}{m}t)\hat i$
de-Broglie wavelength associated with electron at time t is $λ=\frac{h}{mv}$
$⇒=\frac{h}{m[v_0(1+\frac{eE_0}{m}t)]}=\frac{λ_0}{[1+\frac{eE_0}{m}t]}$ $[∵λ_0=\frac{h}{mv_0}]$