Modern Physics
Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Chapter 15, Problem 35P

(a)

To determine

The momenta of the Σ+ and the π+ particles in units of MeV/c.

(a)

Expert Solution
Check Mark

Answer to Problem 35P

The momentum for Σ+ particle is 686 MeV/c and that of π+ particle is 200 MeV/c .

Explanation of Solution

Write the expression for the momentum of Σ+ particle.

  pΣ+=eBrΣ+        (I)

Here, pΣ+ the momentum of Σ+ particle, e is the charge of the electron, B is the magnitude of the magnetic field and rΣ+ is the radius of curvature of the Σ+ particle.

Write the expression for the momentum of π+ particle.

  pπ+=eBrπ+        (II)

Here, pπ+ the momentum of π+ particle and rπ+ is the radius of curvature of the π+ particle.

Conclusion:

The value of e is 1.602177×1019 C .

Substitute 1.602177×1019 C for e , 1.15 T for B and 1.99 m for rΣ+ in equation (I) to find pΣ+ .

  pΣ+=(1.602177×1019 C)(1.15 T)(1.99 m)1 MeV/c5.344288×1022 kgm/s=686.075081 MeV/c686 MeV/c

Substitute 1.602177×1019 C for e , 1.15 T for B and 0.580 m for rπ+ in equation (II) to find pπ+ .

  pπ+=(1.602177×1019 C)(1.15 T)(0.580 m)1 MeV/c5.344288×1022 kgm/s=199.961581 MeV/c200 MeV/c

Therefore, the momentum for Σ+ particle is 686 MeV/c and that of π+ particle is 200 MeV/c .

(b)

To determine

The momentum of the neutron.

(b)

Expert Solution
Check Mark

Answer to Problem 35P

The momentum of the neutron is 627 MeV/c .

Explanation of Solution

Take φ to be the angle made by the neutron’s path with the path of the Σ+ at the moment of the decay.

Write the expression for the conservation of momentum.

  pncosφ+pπ+cosθ=pΣ+

Here, pn is the momentum of the neutron and θ is the angle between the momenta of the Σ+ and the π+ particles at the moment of decay.

Substitute 199.961581 MeV/c for pπ+ , 64.5° for θ and 686.075081 MeV/c for pΣ+ in the above equation.

  pncosφ+(199.961581 MeV/c)cos64.5°=686.075081 MeV/cpncosφ=599.989401 MeV/c        (III)

Write the expression for the conservation of momentum.

  pnsinφ=pπ+sinθ

Substitute 199.961581 MeV/c for pπ+ and 64.5° for θ in the above equation.

  pnsinφ=(199.961581 MeV/c)sin64.5°=180.482380 MeV/c        (IV)

Write the expression for pn .

  pn=(pncosφ)2+(pnsinφ)2

Put equations (III) and (IV) in the above equation.

  pn=(599.989401 MeV/c)2+(180.482380 MeV/c)2=627 MeV/c

Conclusion:

Therefore, the momentum of the neutron is 627 MeV/c .

(c)

To determine

The total energy of the π+ particle, that of neutron and that of Σ+ particle.

(c)

Expert Solution
Check Mark

Answer to Problem 35P

The total energy of the π+ particle is 244 MeV , that of neutron is 1130 MeV and that of Σ+ particle is 1370 MeV .

Explanation of Solution

Write the equation for the total energy of the π+ particle.

  Eπ+=(pπ+c)2+(mπ+c2)2

Here, Eπ+ is the total energy of the π+ particle, mπ+ is the mass the π+ particle and c is the speed of light in vacuum.

Substitute 199.961581 MeV/c for pπ+ and 139.6 MeV for mπ+c2 in the above equation.

  Eπ+=((199.961581 MeV/c)c)2+(139.6 MeV)2=243.870445 MeV244 MeV

Write the equation for the total energy of the neutron.

  En=(pnc)2+(mnc2)2

Here, En is the total energy of the neutron and mn is the mass the neutron.

Substitute 626.547022 MeV/c for pn and 139.6 MeV for mπ+c2 in the above equation.

  En=((626.547022 MeV/c)c)2+(139.6 MeV)2=1129.340219 MeV1130 MeV

Write the equation for the total energy of the Σ+ particle.

  EΣ+=Eπ++En

Here, EΣ+ is the total energy of the Σ+ particle.

Substitute 243.870445 MeV for Eπ+ and 1129.340219 MeV for En in the above equation to find EΣ+ .

  EΣ+=243.870445 MeV+1129.340219 MeV=1373.210664 MeV1370 MeV

Conclusion:

Therefore, the total energy of the π+ particle is 244 MeV , that of neutron is 1130 MeV and that of Σ+ particle is 1370 MeV .

(d)

To determine

The mass of the Σ+ particle.

(d)

Expert Solution
Check Mark

Answer to Problem 35P

The mass of the Σ+ particle is 1190 MeV/c2 .

Explanation of Solution

Write the expression for mΣ+c2 .

  mΣ+c2=EΣ+2+(pΣ+c)2

Here, mΣ+ is the mass the Σ+ particle.

Substitute 1373.210664 MeV for EΣ+ and 686.075081 MeV/c for pΣ+ in the above equation.

  mΣ+c2=(1373.210664 MeV)2+((686.075081 MeV/c)c)2=1190 MeVmΣ+=1190 MeV/c2

Conclusion:

Therefore, the mass of the Σ+ particle is 1190 MeV/c2.

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