Question

What speed must an electron have if its momentum is to be the same as that of an x-ray photon with a wavelength of 0.20 nm?

Answer 1

The momentum of a photon is given by the following relationship:
$p= \frac{h}{\lambda}$
where
h is the Planck constant
$\lambda$ is the photon wavelength

For the photon in our problem, $\lambda=0.20 nm = 0.20 \cdot 10^{-9}m$, so its momentum is
$p= \frac{h}{\lambda}= \frac{6.6 \cdot 10^{-34} Js}{0.20 \cdot 10^{-9} m}=3.3 \cdot 10^{-24} kg m/s$

The electron must have the same momentum of this photon, and its momentum is given by (in the non-relativistic approximation)
$p=mv$
where
m is the electron mass
v is its speed

Re-arranging this formula, we can calculate the electron speed:
$v= \frac{p}{m}= \frac{3.3 \cdot 10^{-24} kg m/s}{9.1 \cdot 10^{-31} kg} =3.6 \cdot 10^6 m/s$

And this velocity is quite small compared to the speed of light ($c=3 \cdot 10^8 m/s$), so the non-relativistic approximation that we used for the electron's momentum is valid.