Ncert Exercises & Solutions

Question 11.34:
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 the early 1970s 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 MeV.)

Hint: $\mathrm{E}^{2}=\mathrm{p}^{2} \mathrm{c}^{2}+\mathrm{m}_{0}^{2} \mathrm{c}^{4 }$

Step 1: Find the momentum of a proton.
The momentum of a proton or a neutron is given as:

$\begin{aligned} p & = \frac{h}{λ} \\ = \frac{6 .6 \times 10^{- 34}}{10^{- 15}} = 6 .6 \times 10^{- 19} kg m / s \end{aligned}$

Step 2: Find the electron energy emitted.
The relativistic relation for energy (E) is given as:

$\begin{aligned} E^{2} & = p^{2} c^{2} + m_{0}^{2} c^{4} \\ = {(6 .6 \times 10^{- 19} \times 3 \times 10^{8})}^{2} + {(0 .8176 \times 10^{- 13})}^{2} \\ = 392 .04 \times 10^{- 22} + 0 .6685 \times 10^{- 36} \\ \approx 392 .04 \times 10^{- 22} \\ ∴ E & = 1 .98 \times 10^{- 10} J \\ = \frac{1 .98 \times 10^{- 10}}{1 .6 \times 10^{- 19}} \\ = 1 .24 \times 10^{9} eV = 1 .24 BeV \end{aligned}$

Thus, the electron energy emitted from the accelerator at Stanford, USA might be of the order of 1.24 BeV.