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Chapter 30: Problem 12

A proton in Fermilab's Tevatron is accelerated through a potential difference of 2.5 MV during each revolution around the ring of radius \(1.0 \mathrm{km} .\) In order to reach an energy of 1 TeV, how many revolutions must the proton make? How far has it traveled?

Short Answer

Expert verified
Answer: The proton travels 2.5 x 10^9 meters to reach an energy of 1 TeV.

Step by step solution

01

Calculate the energy gained by the proton in each revolution

To find the energy gained by the proton in each revolution, we can use the following formula: \(E = qV\). where \(E\) = energy gained by the proton, \(q\) = charge of the proton (\(1.6\times10^{-19} \,\text{C}\)), \(V\) = potential difference (\(2.5\times10^{6}\,\text{V}\)). \(E = (1.6\times10^{-19} \,\text{C})(2.5\times10^{6}\, \text{V}) = 4.0\times10^{-13}\, \text{J}\). So, the proton gains \(4.0\times10^{-13}\,\mathrm{J}\) of energy in each revolution.
02

Calculate the number of revolutions needed to reach 1 TeV

The energy required to reach 1 TeV (tera-electronvolt) is: \(1\,\mathrm{TeV} = 1\times10^{12}\,\mathrm{eV} = 1\times10^{12}\times1.6\times10^{-19}\,\mathrm{J} = 1.6\times10^{-7}\,\mathrm{J}\). Now, we can find the number of revolutions needed using the following formula: Number of revolutions = \(\frac{\text{Total energy required}}{\text{Energy gained per revolution}}\) Number of revolutions = \(\frac{1.6\times10^{-7}\,\text{J}}{4.0\times10^{-13}\, \text{J}} = 4.0\times10^{5}\) So, the proton needs to make \(4.0\times10^{5}\) revolutions to reach 1 TeV.
03

Calculate the distance travelled by the proton

To calculate the distance travelled by the proton, we will use the following formula: Distance travelled = Number of revolutions \(\times\) circumference of the ring First, let's find the circumference of the ring. Circumference = \(2\pi r\) where \(r\) is the radius (\(1.0\,\text{km} = 1000\,\text{m}\)). Circumference = \(2\pi(1000\,\text{m}) \approx 6283.2\,\text{m}\) Now, we can find the distance travelled: Distance travelled = \(4.0\times10^{5} \times 6283.2\,\text{m} = 2.5\times10^{9}\,\text{m}\) The proton travels \(2.5\times10^{9}\,\text{m}\) to reach an energy of 1 TeV.

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