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Chapter 39: Problem 44

(a) What's the relativistic factor \(\gamma\) for a 7 -TeV proton in the Large Hadron Collider? (b) Find an accurate value for the proton's speed.

Short Answer

Expert verified
The relativistic factor \(\gamma\) of a 7-TeV proton in the Large Hadron Collider is approximately 7460. The proton's speed is approximately \(0.999999999997c\) where \(c\) is the speed of light.

Step by step solution

01

Find the Mass of the Proton

The mass of a proton is known. It is approximately \(1.67 × 10^{-27}\) kg.
02

Convert Energy to Joules

In this problem, we are given that the energy is 7 TeV. The conversion 1 TeV = \(1.602 × 10^{-7}\) J can be used to convert the energy from TeV to joules. So, the energy \(E = 7 × 1.602 × 10^{-7}\) J.
03

Calculate the Relativistic Factor

Next, solve for \(\gamma\) in the equation \(E= \gamma m c^{2}\) where \(E\) is the energy and \(m\) is the mass of the proton. Rearranging the equation as \(\gamma = \frac{E}{m c^{2}}\) and plugging in the values of \(E\), \(m\) and \(c\) (the speed of light \(c = 3 × 10^{8}\) m/s), we can find the relativistic factor \(\gamma\).
04

Calculate the Speed of the Proton

Use the relativistic factor to calculate the speed using the equation for \(\gamma\), rearranged as \(v = c \sqrt{1- \frac{1}{\gamma^{2}}}\). This equation can be used to solve for the speed \(v\) of the proton.

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