Ch. 36 - The Special Theory of Relativity

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 36 - The Special Theory of Relativity

Problem 75

Chapter 35, Problem 75

What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 37–5.)

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Identify the rest mass energy of an electron. The rest mass energy (E) can be calculated using the equation E = mc^2, where m is the mass of the electron and c is the speed of light.

Since a positron has the same mass as an electron, its rest mass energy is the same as that of an electron. Therefore, calculate the rest mass energy for the positron using the same formula E = mc^2.

Add the rest mass energies of the electron and the positron to find the total minimum energy required to produce both particles. This is because energy must be conserved, and the creation of these particles requires at least the sum of their individual rest mass energies.

Consider that the process must also conserve momentum. In scenarios where particles are created from energy (like in particle-antiparticle creation), the system's total momentum before and after the creation must be zero if initially there was no momentum. This is typically achieved in laboratory conditions by the particles being emitted in opposite directions.

The minimum amount of electromagnetic energy required is thus twice the rest mass energy of an electron (or a positron), as calculated in the previous steps. This is the threshold energy to ensure both mass and charge conservation laws are satisfied in the creation of an electron-positron pair.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Pair Production

Pair production is a quantum phenomenon where energy is converted into a particle-antiparticle pair, such as an electron and a positron. This process occurs when a photon with sufficient energy interacts with a nucleus or another particle, allowing the energy to be transformed into mass, as described by Einstein's equation E=mc². The minimum energy required for this process is equivalent to the combined rest mass energy of the electron and positron.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In the context of pair production, the energy of the incoming photon must equal the total rest mass energy of the produced particles, ensuring that the energy before and after the interaction remains constant. This principle is fundamental in understanding particle interactions in physics.

Rest Mass Energy

Rest mass energy is the energy equivalent of a particle's mass when it is at rest, calculated using the formula E=mc², where E is energy, m is mass, and c is the speed of light. For an electron and a positron, each has a rest mass of approximately 0.511 MeV/c², so the total rest mass energy required to create both particles is 1.022 MeV. This energy threshold is crucial for initiating pair production.

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