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 {\rm 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 in m
\lambda_{\rm dental}
\lambda_{\rm microtomography}

Answer 1

A)
`5.0\cdot 10^(-11) m`

The energy of an x-ray photon used for single dental x-rays is

`E=25 keV = 25,000 eV \cdot (1.6\cdot 10^(-19) J/eV)=4\cdot 10^(-15) J`

The energy of a photon is related to its wavelength by the equation

`E=(hc)/(\lambda)`

where

`h=6.63\cdot 10^(-34)Js` is the Planck constant

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

`\lambda` is the wavelength

Re-arranging the equation for the wavelength, we find

`\lambda=(hc)/(E)=((6.63\cdot 10^(-34) Js)(3\cdot 10^8 m/s))/(4\cdot 10^(-15)J)=5.0\cdot 10^(-11) m`

B)
`2.0\cdot 10^(-11) m`

The energy of an x-ray photon used in microtomography is 2.5 times greater than the energy of the photon used in part A), so its energy is

`E=2.5 \cdot (4\cdot 10^(-15)J)=1\cdot 10^(-14) J`

And so, by using the same formula we used in part A), we can calculate the corresponding wavelength:

`\lambda=(hc)/(E)=((6.63\cdot 10^(-34) Js)(3\cdot 10^8 m/s))/(1\cdot 10^(-14)J)=2.0\cdot 10^(-11) m`