A photon of energy E strikes an electron at rest undergoes pair production, producing a positive electron(positron) and an electron,
photon+electron produce positron+electron+electron
The two electrons and the positron move off with identical momenta in the direction of the initial photon. Find the kinetic energy of the three final particles and find the energy E of the photon(Hint: conserve momentum and relativistic energy)
To find the kinetic energy of the three final particles and the energy E of the initial photon, we can use the principles of conservation of momentum and relativistic energy.
Let's denote the kinetic energy of the photon as KE_photon, the kinetic energy of the electron as KE_electron, and the kinetic energy of the positron as KE_positron.
Conservation of momentum tells us that the total initial momentum before pair production is equal to the total final momentum after pair production. Since the two electrons and the positron move off with identical momenta in the direction of the initial photon, we can write the momentum conservation equation as:
0 = 2p + p,
where p is the momentum magnitude of the initial photon, and the factor of 2 accounts for the two electrons.
From this equation, we find that p = 0. This means that the initial photon has no momentum, which indicates that it is at rest. Therefore, we can conclude that the energy E of the initial photon is entirely in the form of its rest energy (E = mc^2), where c is the speed of light in a vacuum.
Now, let's consider the conservation of relativistic energy. The total initial energy before pair production is equal to the total final energy after pair production. Since the electron and positron are at rest initially, their total initial energy is equal to the rest energies of the electron and positron, which can be expressed as:
Total initial energy = mc^2 + mc^2 = 2mc^2,
where m is the rest mass of the electron (or positron).
The total final energy after pair production is the sum of the kinetic energies of the three particles:
Total final energy = KE_photon + KE_electron + KE_electron + KE_positron.
Since the two electrons and the positron move off with identical momenta, they all have the same kinetic energy, denoted as KE:
Total final energy = KE_photon + 3KE.
According to conservation of energy, the total initial energy is equal to the total final energy:
2mc^2 = KE_photon + 3KE.
Now, we need to solve the above equation for the values of KE_photon, KE_electron, and KE_positron. We can do this by using known values for the rest mass of the electron (m) and the speed of light (c).
Once we have these values, we can substitute them back in the expressions for KE_photon, KE_electron, and KE_positron, respectively, to find the kinetic energy of each particle.
Note: The exact values will depend on the given values of m and c in the problem, which were not mentioned in the question.