# The wavelength of a probe is roughly a measure of the size of a structure that it can probe in some detail.

**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 MeV.)

**Solution:**

Wavelength of a proton or a neutron, *λ* ≈ 10−15 m

Rest mass energy of an electron:

*m*0*c*2 = 0.511 MeV

= 0.511 × 106 × 1.6 × 10−19

= 0.8176 × 10−13 J

Planck’s constant, *h* = 6.6 × 10−34 Js

Speed of light, *c* = 3 × 108 m/s

The momentum of a proton or a neutron is given as:

$p=\frac{h}{\lambda}$

$=\frac{6.6 \times 10^{-34}}{10^{-15}}=6.6 \times 10^{-19} \mathrm{~kg} \mathrm{~m} / \mathrm{s}$

The relativistic relation for energy (*E*) is given as:

$E^{2}=p^{2} c^{2}+m_{0}^{2} c^{4}$

$=\left(6.6 \times 10^{-19} \times 3 \times 10^{8}\right)^{2}+\left(0.8176 \times 10^{-13}\right)^{2}$

$=392.04 \times 10^{-22}+0.6685 \times 10^{-26}$

$\approx 392.04 \times 10^{-22}$

$\therefore E=1.98 \times 10^{-10} \mathrm{~J}$

$=\frac{1.98 \times 10^{-10}}{1.6 \times 10^{-19}}$

$=1.24 \times 10^{9} \mathrm{eV}=1.24 \mathrm{BeV}$

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

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