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.)

Answer 1

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

Rest mass energy of an electron:

m₀c² = 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" = "h"/lambda`

= `(6.6 xx 10^(-34))/10^(-15)`

= 6.6 × 10⁻¹⁹ kg m/s

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

`"E"^2 = "p"^2"c"^2 + "m"_0^2"c"^4`

= `(6.6 xx 10^(-19) xx 3 xx 10^8)^2 + (0.8176 xx 10^(-13))^2 `

= `392.04 xx 10^(-22) + 0.6685 xx 10^(-26)`

≈ 392.04 × 10⁻²²

∴ E = 1.98 × 10⁻¹⁰ J

= `(1.98 xx 10^(-10))/(1.6 xx 10^(-19))`

= 1.24 × 10⁹ eV

= 1.24 BeV

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