Chapter 32, Problem 51GP

To determine

To Show: The order of magnitude of energy as the order of mass of the ${\mathrm{W}}^{\pm }$ .

Answer to Problem 51GP

The order of magnitude of energy is same as the mass of ${\mathrm{W}}^{\pm }$ .

Explanation of Solution

Given:

Distance, $\mathrm{R}={10}^{-18}\text{m}$

Energy for ${\mathrm{W}}^{\pm }$ = $\mathrm{m}{\mathrm{c}}^2$

Formula used:

Energy -time uncertainty,

  $\mathrm{\Delta }\mathrm{E}\mathrm{\Delta }\mathrm{t}\ge \frac{\mathrm{h}}{4\mathrm{\pi }}$

  $\mathrm{\Delta }\mathrm{t}=\frac{\mathrm{R}}{\mathrm{c}}$

Where, $\mathrm{h}$ = Planck’s constant

  $\mathrm{R}$ = distance

  $\mathrm{c}$ = speed of light

  $\mathrm{E}$ = energy

Calculation:

The minimum energy can be calculated as,

  $\begin{array}{l}{\mathrm{E}}_{\mathrm{n}\mathrm{i}\mathrm{n}}=\frac{\mathrm{h}}{4\mathrm{\pi }\mathrm{\Delta }\mathrm{t}}\\ {\mathrm{E}}_{\mathrm{n}\mathrm{i}\mathrm{n}}=\frac{\mathrm{h}\mathrm{c}}{4\mathrm{\pi }\mathrm{R}}\end{array}$

  $\begin{array}{l}\mathrm{E}=\frac{6.6\times {10}^{-34}\times 3\times {10}^8}{4\mathrm{\pi }\left({10}^{-18}\right)}\text{ J}\\ \mathrm{E}=\frac{6.6\times {10}^{-34}\times 3\times {10}^8}{4\mathrm{\pi }\left({10}^{-18}\right)}\left(\frac{1}{1.6\times {10}^{-19}}\right)\text{ eV}\\ \mathrm{E}=0.9847\times {10}^{11}\text{ eV}\\ \mathrm{E}=98.47\text{ GeV}\end{array}$

Therefore, the energy = $9.847\times {10}^{10}\text{ eV}$

The mass of ${\mathrm{W}}^{\pm }=8\times {10}^{10}{\text{ eV/c}}^{\text{2}}$

Conclusion:

The order of magnitude of energy is same as the mass of ${\mathrm{W}}^{\pm }$ .