Question

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

Short answer

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

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 $783\mathrm{MeV}/{\mathrm{c}}^2$, and the minimum uncertainty energy is:

$\Delta E=783\mathrm{MeV}\left(1.60\times {10}^{-13}\mathrm{J}/\mathrm{MeV}\right)$

$\Delta E=1.25\times {10}^{-10}\mathrm{J}$

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

$\delta t=\frac{h}{2\Delta E}$

$\delta t=\frac{1.055\times {10}^{-34}\mathrm{J}·\mathrm{s}}{2\left(1.25\times {10}^{-10}\mathrm{J}\right)}$

$\delta t=4.21\times {10}^{-25}\mathrm{s}$

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 role="math" localid="1663932465949" $1.26\times {10}^{-16}\mathrm{m}$.