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Chapter 30: Problem 7

The energy at which the fundamental forces are expected to unify is about \(10^{19}\) GeV. Find the mass (in kilograms) of a particle with rest energy \(10^{19} \mathrm{GeV}\).

Short Answer

Expert verified
Answer: The mass of a particle with rest energy \(10^{19} \mathrm{GeV}\) is approximately \(1.783 \times 10^{-28} \: kg\).

Step by step solution

01

Convert the energy from GeV to Joules

To convert the energy given in GeV to Joules, we will use the conversion factor mentioned above. We have the energy \(E = 10^{19} \mathrm{GeV}\), and the conversion factor \(1\: GeV = 1.602176634 \times 10^{-10}\: J\). Multiplying the energy in GeV by the conversion factor gives us the energy in Joules: \(E = 10^{19} \mathrm{GeV} \times 1.602176634 \times 10^{-10} \: J/\mathrm{GeV}\)
02

Calculate the mass using Einstein's equation

Now that we have the energy in Joules, we can use Einstein's equation \(E=mc^2\) to find the mass. Rearrange the equation to solve for the mass: \(m = \frac{E}{c^2}\) Where \(E\) is the energy in Joules (calculated in Step 1) and \(c\) is the speed of light, approximately equal to \(2.998\times10^8 \: \mathrm{m/s}\). Substitute the values and calculate the mass: \(m = \frac{10^{19} \mathrm{GeV} \times 1.602176634 \times 10^{-10} \: J/\mathrm{GeV}}{(2.998\times10^8 \: \mathrm{m/s})^2}\)
03

Evaluate and find the mass in kilograms

Finally, we will evaluate the expression to find the mass in kilograms: \(m = \frac{10^{19} \times 1.602176634 \times 10^{-10}}{(2.998\times10^8)^2} \: kg\) \(m \approx 1.783 \times 10^{-28} \: kg\) So, the mass of a particle with rest energy \(10^{19} \mathrm{GeV}\) is approximately \(1.783 \times 10^{-28} \: kg\).

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Most popular questions from this chapter

Estimate the magnetic field strength required at the LHC to make 7.0-TeV protons travel in a circle of circumference \(27 \mathrm{km}\). Start by deriving an expression, using Newton's second law, for the field strength \(B\) in terms of the particle's momentum \(p,\) its charge \(q,\) and the radius \(r\). Even though derived using classical physics, the expression is relativistically correct. (The estimate will come out much lower than the actual value of 8.33 T. In the LHC, the protons do not travel in a constant magnetic field; they move in straight-line segments between magnets.)
Two factors that can determine the distance over which a force can act are the mass of the exchange particle that carries the force and the Heisenberg uncertainty principle \([\mathrm{Eq} .(28-3)] .\) Assume that the uncertainty in the energy of an exchange particle is given by its rest energy and that the particle travels at nearly the speed of light. What is the range of the weak force carried by the \(\mathbf{Z}\) particle that has a mass of $92 \mathrm{GeV} / \mathrm{c}^{2} ?$ Compare this with the range of the weak force given in Table 30.3
The \(K^{0}\) meson can decay to two pions: \(K^{0} \rightarrow \pi^{+}+\pi^{-}\) The rest energies of the particles are: \(K^{0}=497.7 \mathrm{MeV}\) \(\pi^{+}=\pi^{-}=139.6 \mathrm{MeV} .\) If the \(K^{0}\) is at rest before it decays, what are the kinetic energies of the \(\pi^{+}\) and the \(\pi^{-}\) after the decay?
At the Stanford Linear Accelerator, electrons and positrons collide together at very high energies to create other elementary particles. Suppose an electron and a positron, each with rest energies of \(0.511 \mathrm{MeV},\) collide to create a proton (rest energy \(938 \mathrm{MeV}\) ), an electrically neutral kaon \((498 \mathrm{MeV}),\) and a negatively charged sigma baryon \((1197 \mathrm{MeV}) .\) The reaction can be written as: $$\mathrm{e}^{+}+\mathrm{e}^{-} \rightarrow \mathrm{p}^{+}+\mathrm{K}^{0}+\Sigma^{-}$$ (a) What is the minimum kinetic energy the electron and positron must have to make this reaction go? Assume they each have the same energy. (b) The sigma can decay in the reaction \(\Sigma^{-} \rightarrow \mathrm{n}+\pi^{-}\) with rest energies of \(940 \mathrm{MeV}\) (neutron) and \(140 \mathrm{MeV}\) (pion). What is the kinetic energy of each decay particle if the sigma decays at rest?
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