Question

The wavelength of a probe is roughly a measure of the size of a structure that it can probe in some detail. The quark structure of protons and neutrons appears at the minute length-scale of ${10}^{–15}m$ or less. This structure was first probed in early 1970’𝑠 using high energy electron beams produced by a linear accelerator at Stanford, USA. Guess what might have been the order of energy of these electron beams. (Rest mass energy of electron $=0.511\text{MeV})$

Answer 1

Step:1 Find the rest mass energy of the electron.
Formula used: Rest mass energy $=m_0{c}^2$
Wavelength of a proton or a neutron,
$\lambda \approx {10}^{-15}m$
Planck's constant, $h=6.6\times {10}^{-34}Js$
Speed of light, $c=3\times {10}^{8}m/s$
Rest mass energy of an electron:
$m_0{c}^2=0.511\text{MeV}$
$=0.511\times {10}^6\times .16\times {10}^{-19}$
$=8.176\times {10}^{-14}J$
Step:2 Find the momentum of proton or neutron.
Formula used: $\lambda =\frac{h}{p}$
The momentum of a proton or a neutron is given as:
$p=\frac{h}{\lambda }=\frac{6.6\times {10}^{-34}}{{10}^{-15}}=6.6\times {10}^{-19}\text{kg m/s}$
Step:3 Apply relativistic formula to find energy.
Formula used: $E=\sqrt{p^2{c}^2+m_0^2{c}^4}$
The relativistic relation for energy (E) is given as:
$E=\sqrt{p^2{c}^2+m_0^2{c}^4}$
Substituting the values, we get
$=\sqrt{(6.6\times {10}^{-19}\times 3\times {10}^8{)}^2+(8.176\times {10}^{-14}{)}^2}$
$=\sqrt{3.9204\times {10}^{-20}+6.685\times {10}^{-27}}$
$=\sqrt{3.92\times {10}^{-20}}$
$=1.98\times {10}^{-10}J$
$=\frac{1.98\times {10}^{-10}}{1.6\times {10}^{-19}}eV$
$E=1.24\times {10}^{9}eV$
$E=1.24\text{BeV}$
Thus, the electron energy emitted from the accelerator at Stanford. USA might be of the order of few BeV.