Question

What is the minimum energy of a photon capable of producing an electron- positron pair? What is the wavelength of this photon?

Short answer

Answer: The minimum energy required for a photon to produce an electron-positron pair is \(1.64 \times 10^{-13}\) Joules, and the wavelength of this photon is approximately \(1.21 \times 10^{-12}\) meters.

Step-by-step solution

Obtain the given variables and constants

The mass of an electron/positron is given by \(m_e = 9.11 \times 10^{-31} kg\). The speed of light \(c\) is \(3 \times 10^8 m/s\). Planck's constant \(h\) is \(6.63 \times 10^{-34} Js\). We will use these values to find the minimum energy and wavelength of the photon.

Find the minimum energy of the photon

To create an electron-positron pair, the energy of the photon must be at least equal to the energy of both particles: \(E_{photon} \ge E_{electron} + E_{positron}\) Since both particles have the same mass, we can write: \(E_{photon} \ge 2E_{particle} = 2(mc^2)\) Now we can substitute the given values: \(E_{photon} \ge 2(9.11 \times 10^{-31} kg)(3 \times 10^8 m/s)^2\) \(E_{photon} \ge 1.64 \times 10^{-13} J\) The minimum energy of a photon capable of producing an electron-positron pair is \(1.64 \times 10^{-13}\) Joules.

Find the wavelength of the photon

To find the wavelength of the photon, we need to use Planck's equation: \(E_{photon} = h \nu\) where \(\nu\) is the frequency of the photon, related to its wavelength \(\lambda\) by: \(\nu = \frac{c}{\lambda}\) Substitute this into Planck's equation: \(E_{photon} = h \cdot \frac{c}{\lambda}\) Now solve for \(\lambda\): \(\lambda = \frac{h \cdot c}{E_{photon}}\) Substitute the values: \(\lambda = \frac{(6.63 \times 10^{-34} Js)(3 \times 10^8 m/s)}{1.64 \times 10^{-13} J}\) \(\lambda \approx 1.21 \times 10^{-12} m\) The wavelength of the photon required to produce an electron-positron pair is approximately \(1.21 \times 10^{-12}\) meters.

Key concepts

Photon Energy Calculation

The energy of a photon is a fundamental concept in quantum physics and is crucial to understanding phenomena such as electron-positron pair production. Photon energy refers to the amount of energy carried by a single photon and can be determined using the equation
\( E = h u \),
where \(h\) is Planck's constant and \(u\) is the frequency of the photon. This equation implies that photon energy is directly proportional to its frequency: higher frequency photons carry more energy than lower frequency photons. Calculating photon energy is essential for many applications, including determining the minimum energy needed for reactions such as pair production, understanding electromagnetic spectra, and in medical imaging techniques such as X-rays and gamma rays.

Relativistic Mass-Energy Equivalence

Albert Einstein's famous equation
\( E = mc^2 \)
encapsulates the concept of relativistic mass-energy equivalence, which states that mass and energy are interchangeable. This principle is pivotal in high-energy physics and plays a role in processes like electron-positron pair production. The equation tells us that a certain amount of mass \(m\) is equivalent to a certain amount of energy \(E\) when multiplied by the speed of light \(c\) squared. To produce a matter-antimatter pair (like an electron and a positron), the photon must possess at least the energy equivalent to the combined rest mass of the two particles. This requirement ensures that energy conservation is adhered to in the pair production process.

Planck's Equation

Planck's equation,
\( E = h u \),
is pivotal in quantum mechanics. Named after the physicist Max Planck, this equation links the energy of a photon to its frequency (\(u\)). Planck's constant \(h\) is a fundamental physical constant that sets the scale for quantum effects. When we solve problems related to photon energy, such as in the exercise on electron-positron pair production, Planck's equation is used to connect the quantum world, represented by the photon energy, to classical measurements like frequency and wavelength. Understanding this equation is key to unlocking the behavior of light as quanta and the interaction of light with matter at a subatomic level.

Wavelength and Frequency Relationship

The relationship between wavelength \(\lambda\) and frequency \(u\) of a photon is an essential concept in physics, expressed by the equation
\( u = \frac{c}{\lambda} \),
where \(c\) is the speed of light. This inverse relationship indicates that as the wavelength of a photon decreases, its frequency—and hence its energy—increases, and vice versa. When dealing with phenomena such as the creation of an electron-positron pair, understanding how to relate the energy of a photon to its wavelength using this relationship becomes critical. Since the speed of light is a constant, manipulating the wavelength and frequency is often how varying levels of photon energy are examined and quantified in practical applications, including communication technologies and spectroscopy.