Question

Estimate the range of the force mediated by an \({\omega ^0}\) g meson that has mass \(783\;MeV/{c^2}\) .

Short answer

The range of the force is $1.26\times {10}^{-16}\hspace{0.33em}m$.

Step-by-step solution

Definition of force

The term force may be defined as the product of mass and acceleration.

Determine the range of force

The mass of the${\omega }^0$ particle is$783\hspace{0.33em}MeV/c^2$, and the minimum uncertainty energy is:

role="math" localid="1668318864281" $$\begin{gathered} \Delta E=783\hspace{0.33em}MeV(1.60\times {10}^{-13}\hspace{0.33em}J/MeV) \\ \Delta E=1.25\times {10}^{-10}\hspace{0.33em}J \end{gathered}$$

The lifetime $\delta t$of the particle, according to the Heisenberg uncertainty principle:

$$\begin{gathered} \delta t=\frac{h}{2\mathrm{\Delta }E} \\ \delta t=\frac{1.055\times {10}^{-34}\mathrm{J}\cdot \mathrm{s}}{2\left(1.25\times {10}^{-10}\mathrm{J}\right)} \\ \delta t=4.21\times {10}^{-25}\mathrm{s} \end{gathered}$$

If we consider the speed of light is the average speed of the particle than

$$\begin{gathered} r-c\delta t \\ r=\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)\left(4.21\times {10}^{-25}\mathrm{s}\right) \\ r=1.26\times {10}^{-16}\mathrm{m} \end{gathered}$$

Hence, the range of the force is $1.26\times {10}^{-16}\hspace{0.33em}m$.