89,875,517,873,681.77
J
0.001
kg
90,000,000,000,000
kWh
90,000,000,000,000
MeV
90,000,000,000,000
kt
1
89,875,517,873,681.77
J
0
J
89,875,517,873,681.77
J
0.001
kg
90,000,000,000,000
kWh
90,000,000,000,000
MeV
90,000,000,000,000
kt
1
89,875,517,873,681.77
J
0
J
Einstein's famous equation E = mc² is the most celebrated result in the history of science. Published in 1905 as a consequence of special relativity, it states that mass and energy are equivalent and interconvertible: every kilogram of matter corresponds to approximately 9 × 10¹⁶ joules of energy, and conversely, any form of energy has an equivalent mass. The exact value of c² = 8.98755178736818 × 10¹⁶ m²/s² (using c = 299,792,458 m/s exactly).
The most direct applications of E = mc² are in nuclear physics. When uranium-235 fissions, the total mass of products is slightly less than the original mass — the mass deficit is converted to energy at the rate of c² per kilogram. For a typical fission event, the mass loss is about 0.09% of the uranium mass (the binding energy difference between reactants and products). Complete conversion of 1 gram of matter would yield 9 × 10¹³ J ≈ 21 kilotons of TNT — the yield of the Nagasaki bomb from just 1 gram!
In particle physics, E = mc² takes its most general form as E² = (pc)² + (mc²)², where p is momentum. For a massless photon (m = 0), this reduces to E = pc. For a particle at rest (p = 0), it gives E = mc² (rest energy). For a moving massive particle, the total energy is E = γmc², where γ = 1/√(1-v²/c²) is the Lorentz factor.
The equivalence of mass and energy explains the stability of atomic nuclei: why is the mass of a helium-4 nucleus less than four separate protons and neutrons? Because the binding energy (28.3 MeV) that holds the nucleus together manifests as a mass deficit (Δm = 28.3 MeV/c² = 5.01 × 10⁻²⁹ kg), representing about 0.75% of the helium mass. This is the fundamental origin of all nuclear energy.
Anti-matter annihilation achieves 100% mass-to-energy conversion: when a particle meets its antiparticle (e.g., electron + positron → 2 gamma rays), the entire rest mass converts to photon energy. This is the most energy-efficient energy release known — about 100 times more efficient per unit mass than nuclear fission.
Select the calculation mode: convert mass to energy (E = mc²), energy to equivalent mass (m = E/c²), total relativistic energy including kinetic energy (E = γmc²), or energy from a mass defect in atomic mass units. c = 299,792,458 m/s (exact) is used, giving c² = 8.98755178736818 × 10¹⁶ m²/s².
E = mc² means 1 kg of matter = 9 × 10¹⁶ J = 25 billion kWh — equivalent to the entire world's electricity consumption for about 3 years from just 1 kg. The reason we cannot easily harness this energy is that complete matter-to-energy conversion requires antimatter, and nuclear reactions only convert 0.1% of mass to energy.
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Results
1 gram of matter = 9 × 10^13 J = 21.5 kilotons TNT equivalent. This is the complete conversion yield — actual nuclear weapons use <1% of their fissile mass.
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Results
At v=0.9999c, γ=70.71. Proton total energy = 70.71 × 938.272 MeV = 6,636 MeV. At LHC, protons reach γ ≈ 7,000 giving total energy 6.5 TeV per proton.
Yes. Any form of energy has an equivalent mass via m = E/c². A hot object is very slightly heavier than a cold one (by E_thermal/c²). A compressed spring is slightly heavier than a relaxed one. A charged capacitor is slightly heavier than an uncharged one. These effects are completely negligible at everyday scales — a 1 kg iron bar heated from 0 to 1000°C gains about 5 × 10⁻¹³ kg in mass — but are physically real and in principle measurable.
Einstein's 1905 paper 'Does the Inertia of a Body Depend Upon Its Energy Content?' showed that if a body emits radiation of total energy L, its mass decreases by L/c². The derivation considers a body at rest emitting two photons in opposite directions. Analyzing the kinetic energy in a moving frame using Lorentz transformation, he found that the body's inertia changes by E/c². This was a consequence of the Lorentz transformations and energy-momentum conservation.
In U-235 fission: ~0.085% mass conversion (mass defect / parent mass). D-T fusion: ~0.375% mass conversion. Proton-antiproton annihilation: 100% conversion. Hydrogen burning in stars (p-p chain): ~0.7% conversion (4H → He-4, Δm/4m_H ≈ 0.7%). The Sun converts about 4 million tonnes of mass to energy per second via this process.
Theoretically, antimatter annihilation has the highest possible energy density: ~9 × 10¹⁶ J/kg, making it ideal for interstellar spacecraft propulsion. The exhaust velocity approaches c. The practical problem is producing and storing antimatter: current annual production of antihydrogen at CERN is a few nanograms at a cost of roughly $10⁶⁵/kg. Storage in magnetic bottles is extraordinarily challenging.
m = E/c² = 3,600,000 J / (9 × 10¹⁶ m²/s²) = 4 × 10⁻¹¹ kg = 40 picograms. A kilowatt-hour of electricity is equivalent to the mass of about 40 picograms of matter. Your monthly electricity bill (say, 500 kWh) corresponds to the complete conversion of about 20 nanograms — one-fifth of a microgram of matter.
The mass of every atom is slightly less than the sum of its constituent protons, neutrons, and electrons. This mass deficit Δm, called the mass defect, times c² equals the binding energy: Eb = Δm × c². For hydrogen (1 proton + 1 electron): Eb = 13.6 eV/c² = 2.4 × 10⁻³⁵ kg mass deficit. For iron-56 (most tightly bound nucleus): Eb/nucleon = 8.79 MeV, mass deficit = 0.89% of total mass.
Modern physics distinguishes between rest mass (invariant mass, m₀) which is a constant property of the particle, and the relativistic mass m = γm₀ which increases with velocity. The total energy is E = γm₀c² and momentum p = γm₀v. Most physicists today use 'mass' to mean invariant rest mass and write the energy-momentum relation as E² = p²c² + m₀²c⁴, avoiding the concept of relativistic mass as it can be confusing.
1 u = 1.66053906660 × 10⁻²⁷ kg. Energy: E = mc² = 1.66054 × 10⁻²⁷ × (2.998 × 10⁸)² = 1.4924 × 10⁻¹⁰ J = 931.494 MeV. This is the conversion factor used in nuclear physics: 1 u = 931.494 MeV/c². A neutron (1.008665 u) has rest energy 939.565 MeV.
In principle, any concentrated energy can create matter-antimatter pairs if E > 2m_e c² = 1.022 MeV for electron-positron pairs, or higher thresholds for heavier particles. Near neutron stars, gravitational fields can be so intense that Hawking-like processes might create particle pairs. In the early universe, during the quark epoch, the thermal energy kT > 1 GeV ≈ m_proton c² was sufficient for spontaneous quark creation from the vacuum.
Chemical reactions also obey E = mc², but the energy scales are so small that mass changes are unmeasurable. Burning 1 mole of methane releases 890 kJ = 8.9 × 10⁵ J. Mass change: Δm = 8.9 × 10⁵ / (9 × 10¹⁶) = 9.9 × 10⁻¹² kg = 9.9 nanograms out of 16 grams of methane — a relative mass change of 6 × 10⁻¹³, far below any scale of measurement. Only nuclear reactions (Δm/m ≈ 10⁻³) produce measurable mass changes.
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