Explanation:
Initial de Broglie wavelength of electron,
\({\lambda _0} = \frac{h}{{m{v_0}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
Force on electron in electric field,
\(\overline F = - e\overline E = - e\left[ { - {E_0}\widehat i} \right] = e{E_0}\widehat i\)
Acceleration of electron \(a = \frac{F}{m} = \frac{{e{E_0}}}{{\;m}}\)
Velocity of electron after time \(t\),
\(v = {v_0} + \left( {\frac{{e{E_0}}}{{\;m}}} \right)t\)
\( = \left( {{v_0} + \frac{{e{E_0}}}{{\;m}}t} \right) = {v_0}\left( {1 + \frac{{e{E_0}}}{{m{v_0}}}t} \right)\)
de Broglie wavelength associated with electron at time \(t\) is
\(\lambda = \frac{h}{{m{v_0}}} = \frac{h}{{m\left[ {{v_0}\left( {1 + \frac{{e{E_0}}}{{m{v_0}}}t} \right)} \right]}} = \frac{{{\lambda _0}}}{{m\left[ {1 + \frac{{e{E_0}}}{{m{v_0}}}t} \right]}}\)
[from Eq. (1)]