Chapter 28, Problem 7P

To determine

The uncertainty in the mass of a muon.

Answer to Problem 7P

The answer is $\text{3}\text{.0}{\text{×10}}^{\text{-10}}{\text{eV/c}}^2$

Explanation of Solution

Given info: Mass is $\text{105}\text{.7}{\text{MeV/c}}^2$

Time is $\text{2}\text{.20}\text{μs}\text{= }\text{2}{\text{.2×10}}^{\text{-6}}\text{s}$

Formula used:

It is known that $\text{ΔE }\ge \frac{\text{h}}{\text{Δt}}$

Where,

  $h$ is Planck’s constant, $1.055\times {10}^{-34}J\cdot s$

  $\Delta \text{E}$ is uncertainty in Energy

  $\Delta \text{t}$ is uncertainty in time

Calculation:

Calculate the uncertainty in the energy of the muon from $\text{ΔE }\ge \frac{\text{h}}{\text{Δt}}$

  $\begin{array}{l}\text{=}\frac{\text{1}\text{.055}{\text{×10}}^{\text{-34}}\text{J}\cdot \text{s}}{\text{2}{\text{.2×10}}^{\text{-6}}\text{s}}\\ \text{=4}\text{.8}{\text{×10}}^{\text{-29}}\text{J }\\ \text{= 3}\text{.0}{\text{×10}}^{\text{-10}}\text{eV}\end{array}$

So, the uncertainty in the mass is

  $\text{Δm = }\frac{\text{ΔE}}{{\text{c}}^{\text{2}}}\text{=3}\text{.0}{\text{×10}}^{\text{-10}}{\text{eV/c}}^2$

Conclusion:

The required value is $\text{3}\text{.0}{\text{×10}}^{\text{-10}}{\text{eV/c}}^2$ .