Speed of a High-Energy Electron What s the speed of the electrons in the 50-GeV linac at SLAC? Express the answer as a fraction of the speed of light. ORGANIZE AND PLAN This kinetic energy is much higher than the electron s rest energy (0.511 MeV), so relativity is essential here. Useful

Question

Speed of a High-Energy Electron

What s the speed of the electrons in the 50-GeV linac at SLAC?
Express the answer as a fraction of the speed of light.

ORGANIZE AND PLAN This kinetic energy is much higher than the electron s rest energy (0.511 MeV), so relativity is essential here. Useful relativistic relationships from Chapter 20 are [latex]E=K+E_{0} ext { and }[/latex] [latex]E=\gamma m c^{2} ext {, where the relativistic factor is } \gamma=1 / \sqrt{1-v^{2} / c^{2}} [/latex].

[latex] ext { Known: Electron rest energy } m c^{2}=0.511 MeV ; K=50 GeV [/latex].

Accepted Answer

[latex] ext { Because } K \gg E_{0} ext { in this case, }[/latex] a good approximation is [latex]E=K+E_{0} \approx K . ext { Solving } E=\gamma m c^{2} ext { for the relativistic factor, }[/latex]

[latex]\gamma=\frac{E}{m c^{2}}=\frac{5.0 imes 10^{10} eV }{5.11 imes 10^{5} eV }=9.78 imes 10^{4}[/latex].

[latex] ext { Now solving } \gamma=1 / \sqrt{1-v^{2} / c^{2}} ext { for } v ext {, }[/latex]

[latex]\gamma^{2}=1 /\left(1-v^{2} / c^{2} ight)[/latex].

[latex]v^{2} / c^{2}=1-1 / \gamma^{2}=1-\frac{1}{9.56 imes 10^{9}}=0.999999999895[/latex].

Thus

[latex]v=(0.999999999948) c[/latex].

REFLECT That's amazingly close to the speed of light!

Question 26.8: Speed of a High-Energy Electron What s the speed of the elec...

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