Question 26.5: A Speedy Electron GOAL Compute a total energy and a relativi...
A Speedy Electron
GOAL Compute a total energy and a relativistic kinetic energy.
PROBLEM An electron moves with a speed υ = 0.850c. Find its total energy and kinetic energy in mega electron volts (MeV) and compare the latter to the classical kinetic energy ( 10^{\mathrm{6}} eV = 1 MeV).
STRATEGY Substitute into Equation 26.12 to get the total energy and subtract the rest mass energy to obtain the kinetic energy.
E={\frac{mc^{2}}{\sqrt{1~-~{v^{2}}/c^{2}}}}= [26.12]
Substitute values into Equation 26.12 to obtain the total energy:
E={\frac{m_{e}c^{2}}{\sqrt{1~-~{v^{2}}/c^{2}}}}={\frac{(9.11~\times~10^{-31}\mathrm{~kg})(3.00~\times~10^{8}\,\mathrm{m/s})^{2}}{\sqrt{1~-~(0.850c/c)^{2}}}}
=\;1.56\times10^{-13}~J=(1.56\times10^{-13}~J)\left({\frac{1.00\mathrm{~eV}}{1.60~\times~10^{-19} J}}\right)
= 0.975 MeV
The kinetic energy is obtained by subtracting the rest energy from the total energy:
K E=E-m_{e}c^{2}=0.975\ \mathrm{MeV}-\,0.511\ \mathrm{MeV}= 0.464 ~MeV
Calculate the classical kinetic energy:
K E_{\mathrm{classical}}={\frac{1}{2}}m_{e}v^{2}
={\frac{1}{2}}(9.11\times10^{-31}\,{\mathrm{kg}})(0.850\times3.00\times10^{8}\,{\mathrm{m/s}})^{2}
=2.96\times10^{-14}~J=0.185\;\mathrm{MeV}
REMARKS Notice the large discrepancy between the relativistic kinetic energy and the classical kinetic energy.