Answer : The De Broglie wavelength of an electron is ![1.0025\times 10^{-15}m](https://tex.z-dn.net/?f=1.0025%5Ctimes%2010%5E%7B-15%7Dm)
Explanation :
According to de-Broglie, the expression for wavelength is,
The formula used for kinetic energy is,
![K.E=\frac{1}{2}mv^2](https://tex.z-dn.net/?f=K.E%3D%5Cfrac%7B1%7D%7B2%7Dmv%5E2)
The kinetic energy in terms of momentum will be,
p = m v
![K.E=\frac{1}{2}mv^2=\frac{p^2}{2m}](https://tex.z-dn.net/?f=K.E%3D%5Cfrac%7B1%7D%7B2%7Dmv%5E2%3D%5Cfrac%7Bp%5E2%7D%7B2m%7D)
or,
![p=\sqrt{2mK.E}=mv](https://tex.z-dn.net/?f=p%3D%5Csqrt%7B2mK.E%7D%3Dmv)
As we know that,
![K.E=q\times V](https://tex.z-dn.net/?f=K.E%3Dq%5Ctimes%20V)
where,
'q' is charge of electron
and 'V' is potential difference.
So, ![p=\sqrt{2m\times q\times V}=mv](https://tex.z-dn.net/?f=p%3D%5Csqrt%7B2m%5Ctimes%20q%5Ctimes%20V%7D%3Dmv)
The de Broglie wavelength of the electron will be:
.........(1)
where,
h = Planck's constant = ![6.626\times 10^{-34}Js=6.626\times 10^{-34}kgm^2/s](https://tex.z-dn.net/?f=6.626%5Ctimes%2010%5E%7B-34%7DJs%3D6.626%5Ctimes%2010%5E%7B-34%7Dkgm%5E2%2Fs)
m = mass of electron = ![9.1\times 10^{-31}kg](https://tex.z-dn.net/?f=9.1%5Ctimes%2010%5E%7B-31%7Dkg)
q = charge of electron = ![1.6\times 10^{-9}C](https://tex.z-dn.net/?f=1.6%5Ctimes%2010%5E%7B-9%7DC)
V = potential difference = 150 V
Now put all the given values in equation 1, we get:
![\lambda=\frac{6.626\times 10^{-34}kgm^2/s}{\sqrt{2\times (9.1\times 10^{-31}kg)\times (1.6\times 10^{-9}C)\times (150V)}}](https://tex.z-dn.net/?f=%5Clambda%3D%5Cfrac%7B6.626%5Ctimes%2010%5E%7B-34%7Dkgm%5E2%2Fs%7D%7B%5Csqrt%7B2%5Ctimes%20%289.1%5Ctimes%2010%5E%7B-31%7Dkg%29%5Ctimes%20%281.6%5Ctimes%2010%5E%7B-9%7DC%29%5Ctimes%20%28150V%29%7D%7D)
conversion used : ![(1CV=1J=1kgm^2/s^2)](https://tex.z-dn.net/?f=%281CV%3D1J%3D1kgm%5E2%2Fs%5E2%29)
![\lambda=1.0025\times 10^{-15}m](https://tex.z-dn.net/?f=%5Clambda%3D1.0025%5Ctimes%2010%5E%7B-15%7Dm)
Therefore, the De Broglie wavelength of an electron is ![1.0025\times 10^{-15}m](https://tex.z-dn.net/?f=1.0025%5Ctimes%2010%5E%7B-15%7Dm)