Total Energy and Rest Energy, Mass-energy Equivalence

Binding Energy

• The equivalence of mass and energy becomes apparent when we study the binding energy of systems like atoms and nuclei that are formed from individual particles.

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• The potential energy associated with the force keeping the system together is called the binding energy E_B.

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Total Energy and Rest Energy, Mass-energy Equivalence

We rewrite the energy equation in the form

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The term mc² is called the rest energy and is denoted by E₀.

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This leaves the sum of the kinetic energy and rest energy to be interpreted as the total energy of the particle. The total energy is denoted by E and is given by

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(2.63)

(2.64)

(2.65)

Relationship of Energy and Momentum

We square this result, multiply by c², and rearrange the result.

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We use the equation for γ to express β² and find

Expressing β through γ

Energy and Momentum

The first term on the right-hand side is just E², and the second term is E₀². The last equation becomes

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We rearrange this last equation to find the result we are seeking, a relation between energy and momentum.

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or

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Equation (2.70) is a useful result to relate the total energy of a particle with its momentum. The quantities (E² – p²c²) and m are invariant quantities. Note that when a particle’s velocity is zero and it has no momentum, Equation (2.70) correctly gives E₀ as the particle’s total energy.

(2.71)

(2.70)

Useful formulas

from

and

2.13: Computations in Modern Physics

• We were taught in introductory physics that the international system of units is preferable when doing calculations in science and engineering (“everyday” scales).
• In modern physics a somewhat different set of units is often used, which is more convenient for problems considered in modern physics.
• The smallness of quantities often used in modern physics suggests the need for some new units more practical for smaller scales .

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Units of Work and Energy

• Recall that the work done in accelerating a charge through a potential difference is given by W = qV.
• For a proton, with the charge e = 1.602 × 10⁻¹⁹ C being accelerated across a potential difference of 1 V, the work done on the particle is

W = (1.602 × 10⁻¹⁹C)(1 V) = 1.602 × 10⁻¹⁹ J

The Electron Volt (eV)

• The work done to accelerate the proton across a potential difference of 1 V could also be written as

W = (1 e)(1 V) = 1 eV

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• Thus eV, pronounced “electron volt,” is also a unit of energy. It is related to the SI (Système International) unit joule by the 2 previous equations.

1 eV = 1.602 × 10⁻¹⁹ J

Other Units

1. Rest energy of a particle:�Example: E₀ (proton)

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• Atomic mass unit (amu):

Example: carbon-12

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Mass (¹²C atom)

Mass (¹²C atom)

Binding Energy

The binding energy is the difference between the rest energy of the individual particles and the rest energy of the combined bound system.

The binding energy is the work required to pull the particles out of the bound system into separate, free particles at rest

Conservation of energy

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13.6: Fusion

• Similar to the energy emitted by stars, if two light nuclei fuse together, they also form a nucleus with a larger binding energy per nucleon and energy is released. This reaction is called nuclear fusion.

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• The most energy is released if two isotopes of hydrogen fuse together in the reaction.

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Problem

85,Ch2

Fusion is a clean and efficient energy source

Problem 85,Ch2 (solution)

Two high energy protons hit each other headon

Supercollider at CERN

Aerial photograph representing the Large Hadron Collider, with the border between France and Switzerland indicated by a dashed line. The 27 km LHC tunnel cannot be seen, because it is 45 to 175 meters underground, but it is represented by the large circle

The border between Switzerland and France at the bottom of the slide is not shown

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