E = mc²

What it means

Mass and energy are the same physical quantity, just expressed in different units. The conversion factor is c² ≈ 9 × 10¹⁶ J/kg.

So a 1 kg object at rest has rest energy E₀ = mc² = 9 × 10¹⁶ J — equivalent to ~21.5 megatons of TNT. Nature normally doesn't release this fully — only nuclear reactions and antimatter annihilation tap into it significantly.

Numerical examples

Object	Mass	Rest energy
Electron	9.11 × 10⁻³¹ kg	0.511 MeV
Proton	1.673 × 10⁻²⁷ kg	938.3 MeV
Hydrogen atom	1.674 × 10⁻²⁷ kg	939 MeV
1 atomic mass unit (u)	1.661 × 10⁻²⁷ kg	931.5 MeV
Paperclip (~1 g)	1 × 10⁻³ kg	9 × 10¹³ J ≈ 25 GWh
1 kg	1 kg	9 × 10¹⁶ J ≈ 21.5 kt TNT
Earth's mass	6 × 10²⁴ kg	5.4 × 10⁴¹ J
Sun's mass	2 × 10³⁰ kg	1.8 × 10⁴⁷ J

Mass-energy in reactions

Reaction	Mass converted	Energy released
Chemical (TNT explosion, 1 kg)	~10⁻⁸ kg (negligible)	~5 MJ
Nuclear fission (U-235, 1 kg)	~0.001 kg	~8 × 10¹³ J ≈ 19 kt TNT
Nuclear fusion (D+T, 1 kg)	~0.004 kg	~3 × 10¹⁴ J ≈ 80 kt TNT
Antimatter annihilation (1 kg)	1 kg (100%)	9 × 10¹⁶ J ≈ 21.5 Mt TNT
Sun's hydrogen burning	4 × 10⁹ kg/sec	~3.8 × 10²⁶ W (luminosity)

JavaScript — E = mc² calculations

const c = 3e8;
const c_squared = c * c;  // 9e16
const eV = 1.602e-19;
const u = 1.661e-27;  // atomic mass unit

// Mass to energy
function massToEnergy(mass_kg) { return mass_kg * c_squared; }
function massToEnergy_eV(mass_kg) { return massToEnergy(mass_kg) / eV; }

// Convert atomic mass units to MeV
function uToMeV(mass_u) { return uToJ(mass_u) / eV / 1e6; }
function uToJ(mass_u) { return massToEnergy(mass_u * u); }

console.log(`1 u = ${uToMeV(1).toFixed(2)} MeV`);  // 931.49 MeV (well known)

// Common particles
console.log(`Electron rest: ${massToEnergy_eV(9.11e-31) / 1e6} MeV`);
console.log(`Proton rest: ${(massToEnergy_eV(1.673e-27) / 1e6).toFixed(1)} MeV`);

// Nuclear reaction Q-value (mass deficit)
function qValue(mass_initial_u, mass_final_u) {
  return uToMeV(mass_initial_u - mass_final_u);
}

// Hydrogen → helium fusion
// 4 H + 2 e- → He + 2 ν + photons
const m_H = 1.00794;  // atomic
const m_He = 4.002602;
const Q_fusion = qValue(4 * m_H, m_He);
console.log(`p-p fusion Q: ${Q_fusion.toFixed(2)} MeV`);  // ~26 MeV

// Solar luminosity from fusion
function solarFusionMassLoss(luminosity_W) {
  return luminosity_W / c_squared;
}

console.log(`Sun: ${solarFusionMassLoss(3.828e26).toExponential(2)} kg/s converted`);
// ~4.3 × 10⁹ kg/sec — significant!

// Power → mass equivalent
function massForPower(power_W, time_s) {
  return power_W * time_s / c_squared;
}

// 1 GW for 1 year
console.log(`1 GW for 1 year: ${massForPower(1e9, 365*86400).toFixed(2)} kg of mass`);
// ~0.35 kg of pure mass equivalent

// Annihilation: complete mass-to-energy
function annihilationEnergy(mass_kg) { return massToEnergy(mass_kg); }

// 1 kg matter + 1 kg antimatter
console.log(`1 kg + 1 kg annihilation: ${(annihilationEnergy(2) / 1e15).toFixed(0)} PJ`);
// 180 PJ — roughly 43 megatons of TNT

Where E = mc² matters

• Nuclear power. Reactors, weapons, space propulsion (RTGs).
• Particle physics. Mass and energy interchange in collisions; particle creation.
• Astrophysics. Stars convert mass to energy via fusion; solar energy ultimately E = mc².
• Medical imaging. PET scanners detect 511 keV photons from positron annihilation.
• Cosmology. Total mass-energy of universe; dark energy, dark matter framework.
• Engineering. Some advanced thrusters (ion drive, photon rocket concepts).
• Education. Most famous equation; symbol of modern physics.

Common mistakes

• Thinking everyday objects can release ALL their rest energy. Only antimatter annihilation does. Chemistry releases ~10⁻⁹ of rest energy; fission/fusion ~0.1-1%.
• Confusing mass and weight. E = mc² uses MASS (intrinsic). Weight (force from gravity) is irrelevant.
• Treating it as just for atomic bombs. Applies to all energy/mass conversions, including ordinary chemistry (negligible) and antimatter (complete).
• Believing massless particles have no energy. Photons have energy E = hf and momentum p = E/c. The relation is E² = (pc)² + (mc²)²; for m=0, E = pc.
• Confusing rest energy with total energy. Total = γmc² (includes kinetic). Rest energy = mc² (the v=0 limit).
• Thinking c² is a coincidence. c² appears because of how spacetime structure links energy and mass. Lorentz factor γ has 1-v²/c² in it; algebra gives c² as conversion factor.