A photon strikes an electron of massmthat is initially at rest, creati...
Let us work in the system of units in which c = 1
Now since E2 – p2 is an invariant quantity, we equate this in the initial laboratory frame and in the final frame of particles
In lab frame, the electron is at rest.
∴ Eelectron = m(c = 1) (Take m to be the mass of electron)
∴ Total Energy of the system is E + m = Energy of photon + Energy of electron
E, p are the total energy and momentum of the photon.
∴
Laboratory frame ...(1)
In the final state of particles
E2 – p2 = (m + m + m)2
= (3m)2
= 9m2 ...(2)
Equating for the 2 frames (1) = (2)
∴ (E + m)2 – p2 = 9m2
E2 + m2 + 2Em – p2 = 9m2
since a photon is massless
∴ E = p
∴ m2 + 2Em = 9m2
2Em = 8m2
E = 4m
Now in SI units
E = 4mc2
The correct answer is: 4
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A photon strikes an electron of massmthat is initially at rest, creati...
Given:
- Mass of electron, m
- Electron-positron pair is created when a photon strikes an electron
- The photon is destroyed
- The positron and 2 electrons move off at equal speeds along the initial direction of the photon
To find:
Energy of the photon (in units of mc^2)
Explanation:
When a photon strikes an electron at rest, it can undergo pair production, where an electron-positron pair is created. This process conserves both energy and momentum.
Conservation of Energy:
- Initially, the electron is at rest, so its energy is zero.
- The photon has an energy, E, which is given by E = hf, where h is the Planck's constant and f is the frequency of the photon.
- After the pair production, the positron, electron, and their kinetic energies add up to the initial energy of the photon.
- The positron and electrons move off at equal speeds, so they have equal kinetic energies.
Conservation of Momentum:
- Initially, the electron is at rest, so the total momentum is zero.
- After the pair production, the positron and electrons move off in the same direction with equal speeds, so their momenta add up to zero.
Using Conservation of Energy:
- The energy of the photon, E, is equal to the sum of the energies of the positron and electrons.
- Since the positron and electrons have equal kinetic energies, the energy of the photon is 2 times their kinetic energy.
- Let the kinetic energy of each particle be K. Therefore, the energy of the photon is 2K.
Using Conservation of Momentum:
- The momentum of the positron and electrons add up to zero.
- Since they have equal speeds, their momenta cancel each other out.
- The momentum of the photon is zero initially.
Conclusion:
- The energy of the photon is 2 times the kinetic energy of each particle.
- The momentum of the photon is zero.
- Therefore, the energy of the photon is given by E = 2K.
Answer:
The energy of the photon is 2 times the kinetic energy of each particle. Therefore, the energy of the photon is 4 times the rest mass energy of the electron (4mc^2).