91Ó°ÊÓ

The momentum of a photon can be calculated using the relation: \[ p = \frac{E}{c} \] where \( p \) is the momentum, \( E \) is the energy of the photon, and \( c \) is the speed of light (approximately \(3.00 \times 10^8 \;\mathrm{m/s} \)).

To find the momentum in \( \text{kg} \cdot m/s \), convert the given photon energy into Joules. 1 eV = \( 1.602 \times 10^{-19} \;\text{J} \).(a) For the 10.0 MeV gamma ray: \[ E = 10.0 \times 10^6 \;\text{eV} \times 1.602 \times 10^{-19} \;\text{J/eV} = 1.602 \times 10^{-12} \;\text{J} \](b) For the 25 keV x-ray: \[ E = 25 \times 10^3 \;\text{eV} \times 1.602 \times 10^{-19} \;\text{J/eV} = 4.005 \times 10^{-15} \;\text{J} \](c) For the 1.0 µm infrared photon, use the energy-wavelength relationship: \[ E = \frac{hc}{\lambda} \]Here, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \;\text{J.s}\)), and \( \lambda \) is wavelength (\( 1.0 \mu m = 1.0 \times 10^{-6} \;m\)).\[ E = \frac{(6.626 \times 10^{-34} \;\text{J.s})(3.00 \times 10^8 \;\text{m/s})}{1.0 \times 10^{-6}\;m} = 1.988 \times 10^{-19} \;\text{J} \](d) For the 150 MHz radio-wave photon, convert frequency to energy \[ E = hf \]\( f = 150 \times 10^6 \;\text{Hz} \)\[ E = (6.626 \times 10^{-34} \;\text{J.s})(1.50 \times 10^8 \;\text{Hz}) = 9.939 \times 10^{-26} \;\text{J} \]

Use the photon momentum formula \[ p = \frac{E}{c} \].(a) For the 10.0 MeV gamma ray: \[ p = \frac{1.602 \times 10^{-12} \;\text{J}}{3.00 \times 10^8 \;\text{m/s}} = 5.34 \times 10^{-21} \;\text{kg·m/s}\](b) For the 25 keV x-ray: \[ p = \frac{4.005 \times 10^{-15} \;\text{J}}{3.00 \times 10^8 \;\text{m/s}} = 1.335 \times 10^{-23} \;\text{kg·m/s}\](c) For the 1.0 µm infrared photon: \[ p = \frac{1.988 \times 10^{-19} \;\text{J}}{3.00 \times 10^8 \;\text{m/s}} = 6.63 \times 10^{-28} \;\text{kg·m/s}\](d) For the 150 MHz radio-wave photon: \[ p = \frac{9.939 \times 10^{-26} \;\text{J}}{3.00 \times 10^8 \;\text{m/s}} = 3.313 \times 10^{-34} \;\text{kg·m/s}\]

Since 1 eV/c = 1.7827 x 10^-36 kg·m/s, convert the previously found momentums to eV/c.(a) For the 10.0 MeV gamma ray: \[ p = 5.34 \times 10^{-21} \;\text{kg·m/s} / 1.7827 \times 10^{-36} \;\text{kg·m/s per eV/c} = 2.997 \times 10^{15} \;\text{eV/c} \](b) For the 25 keV x-ray: \[ p = 1.335 \times 10^{-23} \;\text{kg·m/s} / 1.7827 \times 10^{-36} \;\text{kg·m/s per eV/c} = 7.49 \times 10^{12} \;\text{eV/c} \](c) For the 1.0 µm infrared photon: \[ p = 6.63 \times 10^{-28} \;\text{kg·m/s} / 1.7827 \times 10^{-36} \;\text{kg·m/s per eV/c} = 3.718 \times 10^{8} \;\text{eV/c} \](d) For the 150 MHz radio-wave photon: \[ p = 3.313 \times 10^{-34}\; \text{kg·m/s} / 1.7827 \times 10^{-36}\; \text{kg·m/s per eV/c} = 1.859 \times 10^2 \; \text{eV/c} \]