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Chapter 10: Problem 69

In the 1980 s, the term picowave was used to describe food irradiation in order to overcome public resistance by playing on the well-known safety of microwave radiation. Find the energy in \(\mathrm{MeV}\) of a photon having a wavelength of a picometer.

Short Answer

Expert verified
The energy of a photon with a wavelength of one picometer is approximately 1.24 MeV.

Step by step solution

01

Convert the wavelength to meters

Since we are given the wavelength in picometers (pm), we first need to convert it to meters (m). The conversion factor is: 1 pm = 1 × 10^{-12} m So, the wavelength of one picometer is \(1 \times 10^{-12}\) meters.
02

Use the frequency-wavelength relationship to find the frequency

Now that we have the wavelength in meters, we can find the frequency using the formula c = λf. Rearranging this formula to solve for f, we get: f = c / λ The speed of light in a vacuum, c, is approximately \(3 \times 10^{8}\) m/s. Therefore, we can plug in the values for c and λ to calculate the frequency: f = \(\frac{3 \times 10^{8}}{1 \times 10^{-12}}\) Hz f = \(3 \times 10^{20}\) Hz
03

Calculate the energy using the energy-frequency relationship

Now we can use the energy-frequency relationship to calculate the energy of a photon with this frequency: E = hf Here, h is Planck's constant, which is approximately \(6.63 \times 10^{-34}\) Js. Plugging in the values for h and f, we find the energy in Joules: E = \((6.63 \times 10^{-34}) (3 \times 10^{20})\) J E = \(1.99 \times 10^{-13}\) J
04

Convert the energy to MeV

Finally, we need to convert the energy from Joules to MeV. The conversion factor is: 1 MeV = \(1.6 \times 10^{-13}\) J So, we can divide the energy in Joules by the conversion factor to find the energy in MeV: E = \(\frac{1.99 \times 10^{-13}}{1.6 \times 10^{-13}}\) MeV E ≈ 1.24 MeV The energy of a photon with a wavelength of one picometer is approximately 1.24 MeV.

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