Chapter 26, Problem 24Q

To determine

(a)

The rest mass energy of the Boson $\left(\text{W}\right)$.

Answer to Problem 24Q

The rest mass energy of the Boson $\left(\text{W}\right)$ is $80.3\text{GeV}$.

Explanation of Solution

Given:

The mass of the intermediate Boson is $m=85.6\left(\text{mass of proton}\right)$.

Formula used:

The expression of the energy is given by,

$E=m{c}^2$

Here, $E$ is the energy, $m$ is the mass and $c$ is the speed of the light.

Calculation:

The mass of the intermediate boson is calculated as,

$\begin{array}{c}m=85.6\left(\text{mass of proton}\right)\\ =85.6\left(1.67\times {10}^{-27}\text{kg}\right)\\ =1.43\times {10}^{-25}\text{kg}\end{array}$

The rest mass energy of the Boson $\left(\text{W}\right)$ is calculated as,

$\begin{array}{c}E=m{c}^2\\ =\left(1.43\times {10}^{-25}\text{kg}\right){\left(3\times {10}^8\text{m}/\text{s}\right)}^2\\ =1.286\times {10}^{-8}\text{J}\end{array}$

Convert the energy in $\text{GeV}$ as,

$\begin{array}{c}E=1.286\times {10}^{-8}\text{J}\\ =\left(1.286\times {10}^{-8}\text{J}\times \frac{1\text{GeV}}{1.602\times {10}^{-10}\text{J}}\right)\\ =80.3\text{GeV}\end{array}$

Conclusion:

The rest mass energy of the Boson $\left(\text{W}\right)$ is $80.3\text{GeV}$.

To determine

(b)

The threshold temperature for the ${\text{W}}^{+}$ and ${\text{W}}^{-}$.

Answer to Problem 24Q

The threshold temperature for ${\text{W}}^{+}$ and ${\text{W}}^{-}$ is $9.31\times {10}^{14}\text{K}$.

Explanation of Solution

Formula used:

The expression for the relation of energy and temperature is given by,

$T=\frac{E}{k}$

Here, $E$ is the energy and $k$ is the Boltzmann constant.

Calculation:

The threshold temperature for ${\text{W}}^{+}$ and ${\text{W}}^{-}$ is calculated as,

$\begin{array}{l}T=\frac{E}{k}\\ =\frac{80.3\text{GeV}}{1.38\times {10}^{-23}\text{J}/\text{K}}\\ =\frac{\left(80.3\text{GeV}\times \frac{1.60\times {10}^{-10}\text{J}}{1\text{GeV}}\right)}{1.38\times {10}^{-23}\text{J}/\text{K}}\\ =9.31\times {10}^{14}\text{K}\end{array}$

Conclusion:

The threshold temperature for ${\text{W}}^{+}$ and ${\text{W}}^{-}$ is $9.31\times {10}^{14}\text{K}$.

To determine

(c)

The time taken by ${\text{W}}^{+}$ and ${\text{W}}^{-}$ to disappear from universe after the big bang.

Answer to Problem 24Q

The time taken by ${\text{W}}^{+}$ and ${\text{W}}^{-}$ to disappear from universe after the big bang is ${10}^{-12}\text{s}$.

Explanation of Solution

Introduction:

After the big bang, the temperature of the initial universe tends to decreases. All the forces and particles were bound together in the initial universe, but after the decrement in the temperature the particles got separated.

The figure showing the change in the energy of the particles and the temperature of the universe with respect to time from the big bang is shown below:

 

Figure (1)

The time taken from the big bang for ${\text{W}}^{+}$ and ${\text{W}}^{-}$ to disappear corresponding to $80.3\text{GeV}$ of energy and $9.31\times {10}^{14}\text{K}$ of temperature is observed as ${10}^{-12}\text{s}$.

Conclusion:

Therefore, the time taken by ${\text{W}}^{+}$ and ${\text{W}}^{-}$ to disappear from the universe after the big bang is ${10}^{-12}\text{s}$.