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Estimate the range of the force mediated by an \(\omega$$^0\) meson that has mass 783 MeV/\(c$$^2\).

Short Answer

Expert verified
The range of the force is approximately \(5.09 \times 10^{-15}\) meters.

Step by step solution

01

Convert Mass to Energy

The mass of the \(\omega^0\) meson is given as 783 MeV/\(c^2\). In particle physics, it is common to convert mass into energy using \(E = mc^2\), though here the numerical value remains the same because of how it is presented. So, \(E = 783\text{ MeV}\).
02

Calculate the Compton Wavelength

The range of a force mediated by a particle can be estimated using the Compton wavelength formula \(\lambda = \frac{\hbar}{mc}\). First, convert the energy units into SI units:\[ E = 783\text{ MeV} = 783 \times 10^6 \times 1.602 \times 10^{-19} \text{ J} \]Then,\[ \lambda = \frac{6.626 \times 10^{-34} \text{ Js}}{783 \times 10^6 \times 1.602 \times 10^{-19} \text{ J} \cdot 3 \times 10^8 \text{ m/s}} \]Simplify the expression to solve for \(\lambda\).
03

Perform the Calculation

Substituting in the values:\[ \lambda = \frac{6.626 \times 10^{-34}}{3.767466 \times 10^{-10}} \]Perform the calculation to find \(\lambda\):\[ \lambda \approx 5.09 \times 10^{-15} \text{ m} \]This result is the Compton wavelength of the \(\omega^0\) meson, which gives an estimation of the range of the force it mediates.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Force Mediation
In particle physics, the concept of force mediation is fundamental. Forces between particles are not transmitted directly, but rather through intermediary particles known as force mediators or gauge bosons. For instance, in the electromagnetic force, the photon serves as the mediator.
These mediators are often virtual particles that cannot be observed directly. The essence of force mediation is that these particles are exchanged between other particles, thus transmitting the force across space.
The \(\omega^0\) meson, with a mass of 783 MeV/\(c^2\), is such a particle that mediates nuclear forces. By understanding the properties of these mediators, such as their Compton wavelength, physicists estimate the range over which they can effectively transmit force. This illustrates the intricate dance of particles within the quantum realm.
Particle Physics
Particle physics explores the fundamental constituents of matter and the forces of nature. It examines the basic building blocks of the universe such as quarks, leptons, and bosons. At this scale, the particles do not behave in ways that align with our everyday experiences.
Particles like the \(\omega^0\) meson are included in this study. These particles can be quite heavy, with their mass commonly expressed in energy units, such as MeV/\(c^2\), instead of kilograms. This is due to the relationship \(E=mc^2\), which unites mass and energy in physics. When exploring these minute entities, the aim is to demystify everything from nuclear reactions to cosmic events.
Ultimately, particle physics helps us comprehend matter's basic architecture, as well as mysterious concepts like dark matter and antimatter.
Energy Units Conversion
In particle physics, energies are often expressed in electronvolts (eV), specifically mega-electronvolts (MeV). These energy units are more apt for the microscopic scale, where conversion is frequently necessary to consistency with the International System of Units (SI).
For example, the energy equivalent of the mass of the \(\omega^0\) meson is initially given in MeV. To convert this into joules, you multiply by the conversion factor \(1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}\). Thus, the expression \(783 \, \text{MeV} = 783 \times 10^6 \times 1.602 \times 10^{-19} \, \text{J}\) holds.
Energy unit conversions are indispensable in calculating other physical properties, such as wavelengths or forces, providing a bridge between theoretical predictions and practical applications.
Range Estimation
Estimating the range of forces in particle physics involves using the concept of the Compton wavelength. This wavelength represents a length scale at which quantum mechanical effects become significant for a particle of a given mass. \(\)To calculate this for a particle, the formula \(\lambda = \frac{\hbar}{mc}\) is employed, where \(\hbar\) is the reduced Planck's constant, \(m\) is the mass of the particle, and \(c\) is the speed of light.
Applying this to the \(\omega^0\) meson, we first convert its mass from energy units (MeV) to joules, which allows us to find \(\lambda\). This serves to approximate the distance over which the meson can mediate forces between nucleons.
Such estimations are vital in the study of nuclear and subatomic interactions, providing insight into the reach of forces in the particle domain.

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Most popular questions from this chapter

The 2.728-K blackbody radiation has its peak wavelength at 1.062 mm. What was the peak wavelength at \(t = 700,000\) y when the temperature was 3000 K?

Deuterons in a cyclotron travel in a circle with radius 32.0 cm just before emerging from the dees. The frequency of the applied alternating voltage is 9.00 MHz. Find (a) the magnetic field and (b) the kinetic energy and speed of the deuterons upon emergence.

(a) A high-energy beam of alpha particles collides with a stationary helium gas target. What must the total energy of a beam particle be if the available energy in the collision is 16.0 GeV? (b) If the alpha particles instead interact in a colliding-beam experiment, what must the energy of each beam be to produce the same available energy?

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One proposed proton decay is \(p^+ \rightarrow e^+ + \pi^0\), which violates both baryon and lepton number conservation, so the proton lifetime is expected to be very long. Suppose the proton half-life were \(1.0 \times 10^{18} y\). (a) Calculate the energy deposited per kilogram of body tissue (in rad) due to the decay of the protons in your body in one year. Model your body as consisting entirely of water. Only the two protons in the hydrogen atoms in each \(H_2O\) molecule would decay in the manner shown; do you see why? Assume that the \(\pi^0\) decays to two \(\gamma\) rays, that the positron annihilates with an electron, and that all the energy produced in the primary decay and these secondary decays remains in your body. (b) Calculate the equivalent dose (in rem) assuming an RBE of 1.0 for all the radiation products, and compare with the 0.1 rem due to the natural background and the 5.0-rem guideline for industrial workers. Based on your calculation, can the proton lifetime be as short as \(1.0 \times 10^{18} y\)?

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