Mass-Energy Equivalence: E = mc²

Einstein's most famous equation, published in 1905 as a consequence of special relativity:

$E = mc^{2}$

This says that mass and energy are interchangeable — they're the same thing in different forms. A small amount of mass contains an enormous amount of energy because $c^{2} = (3 \times 10^{8})^{2} = 9 \times 10^{16}$ m²/s² is a very large number.

The 1-gram calculation

$E = mc^{2} = (0.001\text{ kg})(9 \times 10^{16}\text{ m}^{2}/\text{s}^{2}) = 9 \times 10^{13}\text{ J} = 90\text{ TJ}$

90 terajoules from a single gram. Compare:

• 1 ton of TNT = 4.2 × 10⁹ J
• 1 gram of matter = 90 × 10¹² J = 21,500 tons of TNT ≈ the Hiroshima bomb

A paperclip's worth of matter, fully converted, equals a nuclear weapon.

In practice, nuclear reactions convert only a tiny fraction of mass to energy: fission converts ~0.1%, fusion ~0.7%. Even so, the energy is millions of times greater than chemical reactions (which convert ~$10^{-10}$% of mass).

Where we see E = mc² in action

• Nuclear reactions: The mass of products is slightly less than reactants. The "missing" mass is the energy released.
• Particle-antiparticle annihilation: An electron meets a positron; both vanish, converting 100% of their mass to gamma-ray energy.
• The sun: Converts 4 million tons of mass to energy every second.
• PET scans: Positrons annihilate with electrons in your body, producing gamma rays that form the medical image.

Common mistakes

• Thinking E = mc² means mass is "destroyed." Mass-energy is conserved. In nuclear reactions, the total energy (rest mass + kinetic + radiation) is constant. What changes is the form: some rest mass converts to kinetic energy and radiation.

Try it in the visualization

Enter a mass and see the equivalent energy. Compare to TNT, coal, and nuclear fuel. The scale comparison bars show just how much more energy mass-energy equivalence unlocks compared to chemical reactions.