On the basis of the de Broglie hypothesis, obtain the expression for the de Broglie wavelength associated with an electron accelerated from rest through a p.d. V.

Explanation

Consider an electron accelerated from rest through a potential difference V. Let `v` be the final speed of the electron. We consider the monrelativistic case, `v lt lt c`, where c is the speed of light in free space.
The kinetic energy acquired by the electron is
`KE=(1)/(2) mv^(2)=(1)/(2m)(mv)^(2)=ev " " ` ...(1)
where e and m are the electronic charge and (nonrelativistic) mass.
`therefore (mv)^(2) =2meV`
The momentu of the electron, `p=mv`
`therefore p^(2) =(mv)^(2)=2meV`
`therefore p=mv=sqrt(2meV) " " ` ...(2)
The de Broglie wavelength `(lambda)` associated with the electron is given by the de Broglie equation
`lambda=(h)/(p) " "` ...(3)
where h is Planck's constant.
From Eqs. (2) and (3),
`lambda =(h)/(sqrt(2meV)) " " `...(4)
which is the required expression.