To find the energy of the gamma-ray photon, we can use the equation for the conservation of energy:
E_photon = E_electron + E_positron
Given that each particle has a kinetic energy of 275 keV, we convert this to MeV:
E_particle = 275 keV = 275 * 10^-6 MeV = 0.275 MeV
Therefore, the energy of the gamma-ray photon is:
E_photon = 2 * E_particle = 2 * 0.275 MeV = 0.55 MeV
So, the energy of the photon is 0.55 MeV.
To find the wavelength of the photon, we can use the equation for the energy of a photon:
E_photon = hc / λ
Where:
E_photon is the energy of the photon,
h is Planck's constant (6.626 × 10^-34 J*s),
c is the speed of light (3 × 10^8 m/s),
and λ is the wavelength of the photon.
Let's convert the energy of the photon from MeV to joules:
E_photon = 0.55 MeV = 0.55 * 1.6 × 10^-13 J = 8.8 × 10^-14 J
Now, we rearrange the equation to solve for the wavelength:
λ = hc / E_photon
Substituting the values:
λ = (6.626 × 10^-34 J*s * 3 × 10^8 m/s) / (8.8 × 10^-14 J)
Calculating:
λ ≈ 2.25 × 10^-6 m
Therefore, the wavelength of the photon is approximately 2.25 × 10^-6 meters.