Chapter 41, Problem 40P

(a)

To determine

The time difference between two neutrinos to cover the distance.

Explanation of Solution

Introduction:

Neutrinos travel at the speed nearly equal to the speed of light. Time required is the distance covered by an object travelling with some speed.

Write the expression for time taken by first neutrino.

  $t_1=\frac{x}{u_1}$

Here, $x$ is the distance covered, $u_1$ is the speed of first neutrino and $t_1$ is the required time.

Write the expression for time taken by second neutrino.

  $t_2=\frac{x}{u_2}$

Here, $u_2$ is the speed of second neutrino and $t_2$ is the required time.

Write the expression for time difference.

  $\Delta t=t_2-t_1$

Substitute $\frac{x}{u_1}$ for $t_1$ and $\frac{x}{u_2}$ for $t_2$ in above expression.

  $\begin{array}{c}\Delta t=\frac{x}{u_2}-\frac{x}{u_1}\\ \Delta t=\frac{x\left(u_1-u_2\right)}{u_1{u}_2}\end{array}$

Since, the speed of neutrino is nearly equal to the speed of light; therefore, $u_1{u}_2\approx c^2$ .

Substitute $c^2$ for $u_1{u}_2$ and $\Delta u$ for $u_1-u_2$ in above expression.

  $\Delta t\approx \frac{x\left(\Delta u\right)}{c^2}$

Here, $\Delta u$ is the difference in speed, $c$ is the speed of light and $\Delta t$ is the time difference.

Conclusion:

Thus, the time difference between two neutrinos to cover the distance is $\Delta t\approx \frac{x\left(\Delta u\right)}{c^2}$ .

(b)

To determine

The ratio of speed of neutrino to the speed of light.

Explanation of Solution

Introduction:

Neutrinos travel at the speed nearly equal to the speed of light. Time required is the distance covered by an object travelling with some speed.

Write the expression for theory of relativity.

  $E=\frac{m{c}^2}{\sqrt{1-{\left(u/c\right)}^2}}$

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

Rearrange the above expression in terms of $u/c$ .

  $\frac{u}{c}={\left(1-{\left(\frac{m{c}^2}{E}\right)}^2\right)}^{1/2}$

Expand the above expression through binomial expansion.

  $\frac{u}{c}=1-\frac{1}{2}{\left(\frac{m{c}^2}{E}\right)}^2$

Conclusion:

Thus, the ratio of speed of neutrino to the speed of light is $\frac{u}{c}=1-\frac{1}{2}{\left(\frac{m{c}^2}{E}\right)}^2$ .

(c)

To determine

The speed difference and time difference between two neutrinos.

Explanation of Solution

Given:

The distance of supernova is $170,000c\cdot \text{y}$ .

The energy of first neutrino is $20\text{MeV}$ .

The energy of first neutrino is $5\text{MeV}$ .

The mass of neutrino is $2.0\text{eV}/c^2$

Formula used:

Write the expression for theory of relativity.

  $E=\frac{m{c}^2}{\sqrt{1-{\left(u/c\right)}^2}}$

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

Rearrange the above expression in terms of $u$ .

  $u=c{\left(1-{\left(\frac{m{c}^2}{E}\right)}^2\right)}^{1/2}$

Expand the above expression through binomial expansion.

  $u=c\left(1-\frac{1}{2}{\left(\frac{m{c}^2}{E}\right)}^2\right)$

Write the expression for speed difference.

  $\begin{array}{c}\Delta u=u_1-u_2\\ \Delta u=c\left(1-\frac{1}{2}{\left(\frac{m{c}^2}{E_1}\right)}^2\right)-c\left(1-\frac{1}{2}{\left(\frac{m{c}^2}{E_2}\right)}^2\right)\end{array}$

Here, $u_1$ is the speed of first neutrino, $u_2$ is the speed of second neutrino, $E_1$ is the energy of first neutrino, $E_2$ is the energy of second neutrino and $\Delta u$ is the difference in speed.

Simplify the above expression.

  $\Delta u=\frac{c{\left(m{c}^2\right)}^2\left(E_1^2-E_2^2\right)}{2E_1^2{E}_2^2}$   ....... (1)

Write the expression for time taken by first neutrino.

  $t_1=\frac{x}{u_1}$

Here, $x$ is the distance covered and $t_1$ is the required time.

Write the expression for time taken by second neutrino.

  $t_2=\frac{x}{u_2}$

Here, $t_2$ is the required time.

Write the expression for time difference.

  $\Delta t=t_2-t_1$

Substitute $\frac{x}{u_1}$ for $t_1$ and $\frac{x}{u_2}$ for $t_2$ in above expression.

  $\begin{array}{c}\Delta t=\frac{x}{u_2}-\frac{x}{u_1}\\ \Delta t=\frac{x\left(u_1-u_2\right)}{u_1{u}_2}\end{array}$

Since, the speed of neutrino is nearly equal to the speed of light; therefore, $u_1{u}_2\approx c^2$ .

Substitute $c^2$ for $u_1{u}_2$ and $\Delta u$ for $u_1-u_2$ in above expression.

  $\Delta t\approx \frac{x\left(\Delta u\right)}{c^2}$   ....... (2)

Here, $\Delta t$ is the time difference.

Calculation:

Substitute $2.0\text{eV}/c^2$ for $m$ , $20\text{MeV}$ for $E_1$ and $5\text{MeV}$ for $E_2$ in equation (1).

  $\begin{array}{c}\Delta u=\frac{c{\left(\left(2.0\text{eV}/c^2\right)c^2\right)}^2\left({\left(20\text{MeV}\right)}^2-{\left(5\text{MeV}\right)}^2\right)}{2{\left(20\text{MeV}\right)}^2{\left(5\text{MeV}\right)}^2}\\ =7.5\times {10}^{-14}c\end{array}$

Substitute $170,000c\cdot \text{y}$ for $x$ and $7.5\times {10}^{-14}c$ for $\Delta u$ in equation (2).

  $\begin{array}{c}\Delta t\approx \frac{\left(170,000c\cdot \text{y}\right)\left(7.5\times {10}^{-14}c\right)}{c^2}\\ =\left(1.275\times {10}^{-8}\text{y}\right)\left(\frac{31.56\text{Ms}}{\text{1}\text{y}}\right)\\ =0.40\text{s}\end{array}$

Conclusion:

The speed difference between two neutrinos is $7.5\times {10}^{-14}c$ ; the time difference between two neutrinos is $0.40\text{s}$ .

(c)

To determine

The speed difference and time difference between two neutrinos.

Explanation of Solution

Given:

The distance of supernova is $170,000c\cdot \text{y}$ .

The energy of first neutrino is $20\text{MeV}$ .

The energy of first neutrino is $5\text{MeV}$ .

The mass of neutrino is $20\text{eV}/c^2$

Formula used:

Write the expression for theory of relativity.

  $E=\frac{m{c}^2}{\sqrt{1-{\left(u/c\right)}^2}}$

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

Rearrange the above expression in terms of $u$ .

  $u=c{\left(1-{\left(\frac{m{c}^2}{E}\right)}^2\right)}^{1/2}$

Expand the above expression through binomial expansion.

  $u=c\left(1-\frac{1}{2}{\left(\frac{m{c}^2}{E}\right)}^2\right)$

Write the expression for speed difference.

  $\begin{array}{c}\Delta u=u_1-u_2\\ \Delta u=c\left(1-\frac{1}{2}{\left(\frac{m{c}^2}{E_1}\right)}^2\right)-c\left(1-\frac{1}{2}{\left(\frac{m{c}^2}{E_2}\right)}^2\right)\end{array}$

Here, $u_1$ is the speed of first neutrino, $u_2$ is the speed of second neutrino, $E_1$ is the energy of first neutrino, $E_2$ is the energy of second neutrino and $\Delta u$ is the difference in speed.

Simplify the above expression.

  $\Delta u=\frac{c{\left(m{c}^2\right)}^2\left(E_1^2-E_2^2\right)}{2E_1^2{E}_2^2}$   ....... (1)

Write the expression for time taken by first neutrino.

  $t_1=\frac{x}{u_1}$

Here, $x$ is the distance covered and $t_1$ is the required time.

Write the expression for time taken by second neutrino.

  $t_2=\frac{x}{u_2}$

Here, $t_2$ is the required time.

Write the expression for time difference.

  $\Delta t=t_2-t_1$

Substitute $\frac{x}{u_1}$ for $t_1$ and $\frac{x}{u_2}$ for $t_2$ in above expression.

  $\begin{array}{c}\Delta t=\frac{x}{u_2}-\frac{x}{u_1}\\ \Delta t=\frac{x\left(u_1-u_2\right)}{u_1{u}_2}\end{array}$

Since, the speed of neutrino is nearly equal to the speed of light; therefore, $u_1{u}_2\approx c^2$ .

Substitute $c^2$ for $u_1{u}_2$ and $\Delta u$ for $u_1-u_2$ in above expression.

  $\Delta t\approx \frac{x\left(\Delta u\right)}{c^2}$   ....... (2)

Here, $\Delta t$ is the time difference.

Calculation:

Substitute $20\text{eV}/c^2$ for $m$ , $20\text{MeV}$ for $E_1$ and $5\text{MeV}$ for $E_2$ in equation (1).

  $\begin{array}{c}\Delta u=\frac{c{\left(\left(20\text{eV}/c^2\right)c^2\right)}^2\left({\left(20\text{MeV}\right)}^2-{\left(5\text{MeV}\right)}^2\right)}{2{\left(20\text{MeV}\right)}^2{\left(5\text{MeV}\right)}^2}\\ =7.5\times {10}^{-12}c\end{array}$

Substitute $170,000c\cdot \text{y}$ for $x$ and $7.5\times {10}^{-12}c$ for $\Delta u$ in equation (2).

  $\begin{array}{c}\Delta t\approx \frac{\left(170,000c\cdot \text{y}\right)\left(7.5\times {10}^{-12}c\right)}{c^2}\\ =\left(1.275\times {10}^{-6}\text{y}\right)\left(\frac{31.56\text{Ms}}{\text{1}\text{y}}\right)\\ =40\text{s}\end{array}$

Conclusion:

The speed difference between two neutrinos is $7.5\times {10}^{-12}c$ ; the time difference between two neutrinos is $40\text{s}$ .