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Chapter 31, Problem 28P

(a)

To determine

The magnitude of momentum the particle Σ+ and π+.

(a)

Expert Solution
Check Mark

Answer to Problem 28P

The magnitude of momentum the particle Σ+ and π+ is 686MeV/c_ and 200MeV/c_.

Explanation of Solution

Write the expression for the momentum.

p=mv        (I)

Here, p is the momentum, m is the mass, v is the particle speed.

Write the expression for the momentum for a circular motion by using equation (I).

qvB=mv2rmv=qBrp=qBr        (II)

Here, q is the charge, B is the magnetic field and r is the radius of the path.

Conclusion:

For the particle Σ+

Substitute, 1.602×1019C for q, 1.15T for B, 1.99m for r in equation (II) to find p.

p=[(1.602×1019C)(1.15T)(1.99m)](1.871×1021MeV/ckg-m/s)=686MeV/c

For the particle π+

Substitute, 1.602×1019C for q, 1.15T for B, 0.580m for r in equation (II) to find p.

p=[(1.602×1019C)(1.15T)(0.580m)](1.871×1021MeV/ckg-m/s)=200MeV/c

Thus, the magnitude of momentum the particle Σ+ and π+ is 686MeV/c_ and 200MeV/c_.

(b)

To determine

The magnitude of the momentum of neutron.

(b)

Expert Solution
Check Mark

Answer to Problem 28P

The magnitude of the momentum of neutron is 626MeV/c_.

Explanation of Solution

Write the expression for the magnitude of the momentum of the neutron.

pn=(pncosϕ)2+(pnsinϕ)2        (III)

Here, pn is the magnitude of the momentum of neutron, pncosϕ is the x-component of momentum and pnsinϕ is y-component of momentum.

Write the expression for the x-component of neutron momentum from the conservation of momentum along parallel to original momentum

pΣ+=pncosϕ+pπ+cosθpncosϕ=pΣ+pπ+cosθ        (IV)

Write the expression for the y-component of neutron momentum from the conservation of momentum along perpendicular to original momentum

pnsinϕ=pπ+cosθ        (V)

Write the expression from (I) by using (IV) and (V).

pn=(pΣ+pπ+cosθ)2+(pπ+sinθ)2        (VI)

Conclusion:

Substitute, (686MeV/c) for pΣ+, (200MeV/c) for pπ+, 64.5° for θ in equation (VI) to find pn.

pn=((686MeV/c)[(200MeV/c)cos(64.5°)])2+((200MeV/c)sin(64.5°))2=626MeV/c

Thus, the magnitude of the momentum of neutron is 626MeV/c_.

(c)

To determine

The total energy of the π+ and neutron.

(c)

Expert Solution
Check Mark

Answer to Problem 28P

The total energy of the π+ and neutron is 244MeV_ and 1.13GeV_.

Explanation of Solution

Write the expression for the total energy of π+.

Eπ+=(pπ+c)2+(mπ+c2)2        (VII)

Here, Eπ+ is the total energy of , π+ c is the light of speed, mπ+ is the mass of π+, pπ+ is the momentum of π+.

Write the expression for the total energy of neutron.

En=(pnc)2+(mnc2)2        (VIII)

Here, En is the total energy of neutron, c is the light of speed, mn is the mass of neutron, pn is the momentum of neutron.

Conclusion:

Substitute, 200MeV for pπ+c, 139.6MeV for mπ+c2 in equation (VII) to find Eπ+.

Eπ+=(200MeV)2+(139.6MeV)2=244MeV

Substitute, 626MeV for pnc, 939.6MeV for mnc2 in equation (VIII) to find En.

En=(626MeV)2+(939.6MeV)2=1129MeV=1.13GeV

Thus, the total energy of the π+ and neutron is 244MeV_ and 1.13GeV_.

(d)

To determine

The total energy of the Σ+.

(d)

Expert Solution
Check Mark

Answer to Problem 28P

The total energy of the Σ+ is 1.37GeV_.

Explanation of Solution

Write the expression for the total energy of Σ+.

EΣ+=Eπ++En        (IX)

Here, EΣ+ is the total energy of Σ+.

Conclusion:

Substitute, 244MeV for Eπ+, 1129MeV for En in equation (IX) to find EΣ+.

EΣ+=(244MeV)+(1129MeV)=1373MeV=1.37GeV

Thus, The total energy of the Σ+ is 1.37GeV_.

(e)

To determine

The mass of Σ+.

(e)

Expert Solution
Check Mark

Answer to Problem 28P

The mass of Σ+ is 1.19GeV/c2.

Explanation of Solution

Write the expression for the mass of Σ+ by using rest mass energy.

mΣ+=1c2[EΣ+2(pΣ+c)2]        (X)

Here, mΣ+ is the mass.

Conclusion:

Substitute, 1373MeV for EΣ+, 686MeV for pΣ+c in equation (X) to find mΣ+.

mΣ+=1c2[(1373MeV)2(686MeV)2]=1189MeV/c2=1.19GeV/c2

Thus, the mass of Σ+ is 1.19GeV/c2.

(f)

To determine

Compare the calculated and given value.

(f)

Expert Solution
Check Mark

Answer to Problem 28P

The mass of Σ+ is 1.19GeV/c2.

Explanation of Solution

Write the expression for the comparison between the given and the calculated value of mass.

[(1.19×103MeV/c2)(1189.4MeV/c2)(1189.4MeV/c2)]×100%=0.0504%

The calculated value differs from 0.05% from given value.

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Chapter 31 Solutions

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