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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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