Question

The photon energies used in different types of medical x-ray imaging vary widely, depending upon the application. Single dental x rays use photons with energies of about 25 keV. The energies used for x-ray microtomography, a process that allows repeated imaging in single planes at varying depths within the sample, is 2.5 times greater.
What are the wavelengths of the x rays used for these two purposes?

Answer 1

1. Single dental x-rays: $5.0\cdot 10^{-11}m$

The energy of the photon is

$E=25 keV = 25,000 eV$

Using the conversion factor

$1 eV=1.6\cdot 10^{-19} J$

we can convert it into Joules:

$E=(25,000 eV)(1.6\cdot 10^{-19}J/eV)=4\cdot 10^{-15} J$

The relationship between photon energy and wavelength is

$\lambda=\frac{hc}{E}$

where

$h=6.63\cdot 10^{-34} Js$ is the Planck constant

$c=3\cdot 10^8 m/s$ is the speed of light

E is the energy

Substituting into the formula, we find

$\lambda=\frac{(6.63\cdot 10^{-34}Js)(3\cdot 10^8 m/s)}{4\cdot 10^{-15} J}=5.0\cdot 10^{-11}m$

2. Microtomography: $2.0\cdot 10^{-11} m$

The energy of these photons is 2.5 times greater, so

$E=(2.5)(4\cdot 10^{-15} J)=1\cdot 10^{-14} J$

And by applying the same formula used at point 1, we find the corresponding wavelength:

$\lambda=\frac{(6.63\cdot 10^{-34}Js)(3\cdot 10^8 m/s)}{1\cdot 10^{-14} J}=2.0\cdot 10^{-11}m$