Chapter 30, Problem 78A

(a)

To determine

Time taken by proton to complete one revolution.

Answer to Problem 78A

  $t=2.09\times {10}^{-5}\text{ s}$

Explanation of Solution

Given:

Diameter $d=2$ km $=2000$ m

Radius $r=1000$ m

Speed $v=c=3\times {10}^8$ m/s

Formula used:

Speed = $\frac{\text{Distance}}{\text{time}}$

Calculation:

Time taken by proton = $\frac{\text{Distance travel by proton (circumference of circle)}}{\text{Speed}}$

  $\begin{array}{l}t=\frac{2\pi r}{v}\\ t=\frac{2\pi \times 1000}{3\times {10}^8}\\ t=2.09\times {10}^{-5}\text{ s}\end{array}$

Conclusion:

Hence, the time taken by proton to complete one revolution is $2.09\times {10}^{-5}$ s.

(b)

To determine

The number of revolution to reach 4 GeV energy.

Answer to Problem 78A

  $n=156800$

Explanation of Solution

Given:

Initial energy $E_i=8$ GeV $=8\times {10}^9$ eV

Gain energy in one revolution $E_g=2.5$ MeV $=25\times {10}^5$ eV

Final energy $E_f=400$ GeV $=400\times {10}^9$ eV

Calculation:

Energy difference between initial and final energy,

  $\begin{array}{l}E=E_f-E_i\\ E=400\times {10}^9-8\times {10}^9\\ E=392\times {10}^9\text{ eV}\end{array}$

Now,

  $25\times {10}^5$ eV energy gained = 1 revolution

  $392\times {10}^9$ eV energy gained = $392\times {10}^9\times \frac{1}{25\times {10}^5}$ revolution

  $=156800$ revolution

Conclusion:

Hence, the proton will take 156800 revolutions to reach 4 GeV energy.

(c)

To determine

Time taken by proton to reach 400 GeV energy.

Answer to Problem 78A

  $T=3.277\text{ s}$

Explanation of Solution

Given:

Time taken by proton to complete one revolution $t=2.09\times {10}^{-5}$ s

Number of revolutions $n=156800$

Calculation:

Time taken by proton to reach 400 GeV energy,

  $\begin{array}{l}T=nt\\ T=156800\times 2.09\times {10}^{-5}\\ T=3.277\text{ s}\end{array}$

Conclusion:

Hence, the time taken by proton to reach 400 GeV energy is 3.277 s.

(d)

To determine

The distance travel by proton during this acceleration.

Answer to Problem 78A

  $S=9.831\times {10}^8\text{ m}$

Explanation of Solution

Given:

Total time $T=3.277$ s

Speed $v=c=3\times {10}^8$ m/s

Formula used:

Speed = $\frac{\text{Distance}}{\text{time}}$

Calculation:

Total distance = Speed × Total time

  $\begin{array}{l}S=vT\\ S=3\times {10}^8\times 3.277\\ S=9.831\times {10}^8\text{ m}\end{array}$

Conclusion:

Hence, the distance travel by proton during this acceleration is $9.831\times {10}^8\text{ m}$ .