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# 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 the order of energy of these electron beams. (Take wavelength of a proton = $10^{-15}m$ ; rest mass of energy of electron = $0.511\: MeV$)

$\begin {array} {1 1} (a)\;1.24\: BeV & \quad (b)\;12.4\: BeV \\ (c)\;1.98\: BeV & \quad (d)\;19.8\: BeV \end {array}$

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A)
Ans : (a)
$\lambda = 10^{-15}m$
Rest mass energy of electron: $m_oc^2 = 0.511\: MeV$
$= 0.511 \times 10^6 \times 1.6 \times 10^{-19}$
$= 0.8176 \times 10^{-13} J$
Momentum of a proton, $p = \large\frac{h}{\lambda}$
$= \large\frac{6.6 \times 10^{-34}}{ 10^{-15}} = 6.6 \times 10^{-19} kg m/s$
Relativistic relation for energy (E) is :
$E^2 = p^2c^2 + m^2_oc^4$
$= (6.6 \times 10^{-19} \times 3 \times 10^8 )^2 + (0.8176 \times 10^{-13} )^2$
$= 392.04 \times 10^{-22} + 0.6685 \times 10^{-26}$
$≈ 392.04 \times 10^{-22}$
So, $E = \large\frac{1.98 \times 10^{-10} J}{ 1.6 \times 10^{-19}} = 1.24 \times 10^9 \: eV = 1.24\: BeV$