Relativistic Energy & Mass-Energy Equivalence

The most famous equation in physics, $E=mc^2$, reveals that mass and energy are equivalent. Even a particle at rest has energy $m c^2$. Nuclear reactions convert tiny amounts of mass into enormous energy, and pair production converts pure energy into matter.

Key Concepts

Rest Energy

E_0=mc^2

. A particle at rest has energy

mc^2

purely from its mass. For a proton:

E_0=938.3

MeV; for an electron:

E_0=0.511

MeV.

Total Energy

E=\gamma mc^2

. This includes both rest energy and kinetic energy:

E=E_0+K=(mc^2)+(\gamma-1)mc^2

Mass-Energy Equivalence

\Delta E=\Delta mc^2

. Mass can be converted to energy and vice versa. In nuclear fission, ~0.1% of mass is converted; in fusion, ~0.7%.

Pair Production & Annihilation

A photon with

E\ge2m_ec^2=1.022

MeV can produce an electron-positron pair (in the field of a nucleus). Conversely, an electron and positron annihilate to produce two photons of 0.511 MeV each.

Key Equations

Mass-energy equivalence

E_0 = mc^2

Rest energy; mass and energy are interconvertible at rate c².

Total relativistic energy

E = \gamma mc^2 = \sqrt{(pc)^2 + (mc^2)^2}

Includes rest energy and kinetic energy.

Binding energy

B = [Zm_p + Nm_n - M(Z,N)]c^2

Energy released when Z protons and N neutrons form a nucleus of mass M.

Worked Example

Energy Released in Nuclear Fission

Problem

When U-235 undergoes fission, approximately 0.090% of its mass is converted to energy. A nuclear plant uses 1.0 kg of U-235 per day. How much energy (in J) is released?

Solution

Mass converted: $\Delta m=0.00090\times1.0\text{ kg}=9.0\times10^{-4}$ kg.

E=\Delta m\cdot c^2 = 9.0\times10^{-4}\times(3\times10^8)^2

E = 9.0\times10^{-4}\times9\times10^{16} = 8.1\times10^{13}\text{ J}

That is 81 terajoules — enough to power a city for several days.

Answer

E=8.1\times10^{13}

Practice

Exercises

7 problems

1 of 7

Find the rest energy $E_0$ (in J) of a 1.0 kg object. ( $c=3\times10^8$ m/s)

Your answer

2 of 7

An electron ( $m_ec^2=0.511$ MeV) and a positron annihilate. Each photon produced has energy $E_{\gamma}$ (in MeV). (Both at rest.)

Your answer

MeV

3 of 7

Pair production threshold: minimum photon energy to create an electron-positron pair (in MeV). $m_ec^2=0.511$ MeV.

Unlock Exercise 3