Mass-Energy Equivalence

The equation

Einstein's famous equation states that mass and energy are equivalent:

$E = mc^2$

A body of mass $m$ at rest contains energy $E = mc^2$, where $c = 3 \times 10^8$ m/s is the speed of light.

What it means

Mass is a form of energy. The two are interconvertible. A small amount of mass corresponds to an enormous amount of energy, because $c^2 = 9 \times 10^{16}$ m²/s² is a huge number.

1 kg of mass $\equiv$ $9 \times 10^{16}$ J of energy

For comparison: the Hiroshima bomb released about $6 \times 10^{13}$ J — from just about 1 gram of mass converted to energy.

Origin — from special relativity

Einstein derived this in 1905 from the theory of special relativity. The full energy-momentum relation is:

$E^2 = (mc^2)^2 + (pc)^2$

where $p$ is the relativistic momentum. For a body at rest ($p = 0$):

$E = mc^2$

For a slowly moving body ($v \ll c$), the full expression expands to:

$E \approx mc^2 + \frac{1}{2}mv^2 + \cdots$

The first term $mc^2$ is the rest energy. The second term $\frac{1}{2}mv^2$ is the classical kinetic energy. This shows that classical KE is a small correction to the rest energy for everyday speeds.

Where it matters

Nuclear reactions: In fission and fusion, a small fraction of mass converts to energy. The mass of products is slightly less than the mass of reactants — the "mass defect" multiplied by $c^2$ gives the energy released.

$\Delta E = \Delta m \cdot c^2$

Pair production: A photon with enough energy can spontaneously create an electron-positron pair from pure energy, with mass created from energy.

Particle physics: Particle accelerators convert kinetic energy into mass — creating new particles from collisions.

Mass defect in nuclei

A nucleus is always lighter than the sum of its constituent protons and neutrons. This mass defect $\Delta m$ corresponds to the binding energy:

$E_{binding} = \Delta m \cdot c^2$

This binding energy holds the nucleus together. To break a nucleus apart, you must supply this energy.