E=mc² Calculator

About E=mc² Calculator

The E = mc² Formula

Energy equals mass times the speed of light squared: E = mc², where c = 299,792,458 m/s. Because c² is enormous (about 9 x 10¹⁶ m²/s²), even a tiny mass contains a staggering amount of energy.

Mass	Energy Equivalent	Comparison
1 kg	89.88 petajoules (PJ)	About 21.5 megatons of TNT
1 g	89.88 terajoules (TJ)	About 21.5 kilotons of TNT (similar to Hiroshima)
1 mg	89.88 gigajoules (GJ)	Energy in about 2,500 litres of petrol
1 microgram	89.88 megajoules (MJ)	About 25 kWh of electricity
1 amu (atomic mass unit)	931.5 MeV	Standard in nuclear physics

Worked example: the Hiroshima bomb released about 63 TJ of energy. Using E = mc², the mass converted was m = E/c² = 63 x 10¹² / (9 x 10¹⁶) = 0.0007 kg, or about 0.7 grams. Out of the 64 kg of uranium in the bomb, less than 1 gram was actually converted to energy.

Where Does the Energy Come From?

Einstein's equation does not mean you can extract all the energy from any piece of matter. It describes the maximum energy contained in mass itself. In practice, only nuclear reactions convert measurable amounts of mass to energy.

Process	Mass Converted to Energy	Efficiency (% of mc²)
Chemical combustion (burning petrol)	~0.00000001% of fuel mass	About 10⁻⁸ %
Nuclear fission (uranium-235)	~0.09% of fuel mass	0.09%
Nuclear fusion (hydrogen to helium)	~0.7% of fuel mass	0.7%
Matter-antimatter annihilation	100% of both masses	100%

Nuclear fusion is about 70,000 times more mass-efficient than chemical combustion. This is why the Sun, which runs on fusion, can shine for billions of years while a coal fire burns out in hours.

Mass-Energy in Nuclear Reactions

In nuclear physics, the mass of products is slightly less than the mass of reactants. This "mass defect" is released as energy. For example, in the fusion of deuterium and tritium: D + T produces He-4 + neutron + 17.6 MeV.

Reaction	Energy Released per Event	Mass Converted
U-235 fission	~200 MeV	~0.215 amu (3.57 x 10⁻²⁸ kg)
D-T fusion	~17.6 MeV	~0.0189 amu
Proton-proton chain (Sun)	~26.7 MeV per He-4	~0.0287 amu
Electron-positron annihilation	1.022 MeV	2 x 0.000549 amu (both particles)

Particle Physics and Rest Mass Energy

In particle physics, masses are often expressed in energy units (MeV/c² or GeV/c²) because of the direct mass-energy relationship.

Particle	Rest Mass	Rest Energy (MeV)
Electron	9.109 x 10⁻³¹ kg	0.511
Proton	1.673 x 10⁻²⁷ kg	938.3
Neutron	1.675 x 10⁻²⁷ kg	939.6
Higgs boson	2.23 x 10⁻²⁵ kg	125,100 (125.1 GeV)

The neutron is slightly heavier than the proton (by 1.3 MeV), which is why free neutrons decay into protons with a half-life of about 10 minutes, but protons are stable.

The Full Equation: E² = (pc)² + (mc²)²

E = mc² is the simplified form for objects at rest, and the full relativistic energy-momentum relation is E² = (pc)² + (mc²)², where p is momentum. For a photon (zero rest mass), this collapses to E = pc, which is why light carries energy and momentum despite being massless. For a massive object at rest (p = 0), it reduces to E = mc². At everyday speeds, kinetic energy is well-approximated by the Newtonian 1/2 mv², but at speeds above about 10% of c, the relativistic form becomes essential. The total energy of a moving object is E = γmc², where γ = 1/√(1 - v²/c²) is the Lorentz factor. Subtracting the rest energy gives the relativistic kinetic energy KE = (γ - 1)mc².

How Did Einstein Derive E = mc²?

Einstein derived mass-energy equivalence in a short 1905 follow-up paper to his special relativity paper, titled "Does the Inertia of a Body Depend Upon Its Energy Content?" (Annalen der Physik, Vol. 18). He imagined a body emitting two equal light pulses in opposite directions. By computing the emission in two different reference frames and applying conservation of energy and momentum, he found that the body must lose kinetic energy equal to L/c², where L is the total energy radiated. The only consistent interpretation was that the body lost mass m = L/c² - or equivalently, energy equals mc². The equation therefore applies to any form of internal energy: thermal, chemical, nuclear, or electromagnetic. A hot cup of coffee is very slightly heavier than the same cup cold, although the difference is far too small to measure directly.

Real-World Verification

Mass-energy equivalence has been measured in the laboratory to extreme precision. A 2005 MIT and NIST experiment (Rainville et al., published in Nature, December 2005) tested E = mc² directly by weighing silicon and sulfur nuclei before and after neutron capture, and comparing the mass difference to the gamma-ray energy released. The agreement with E = mc² was confirmed to within 0.00004% - making it one of the most precisely tested equations in physics. Particle accelerators also demonstrate the equation millions of times per second: when protons collide at CERN's Large Hadron Collider at roughly 13 TeV of total energy, the kinetic energy is converted into mass, producing new particles (including the Higgs boson at 125 GeV) that weigh more than the original protons.

Practical Implications and Common Misconceptions

The equation is often misread as saying mass can be "turned into" energy freely, but that is not what it means. Mass and energy are the same quantity measured in different units, related by the conversion factor c². Nothing in a reaction is "destroyed" - the mass is simply carried away as kinetic energy of the products. Some key corrections to common misconceptions:

• Chemical reactions also convert mass, just tiny amounts. Burning 1 kg of petrol releases about 45 MJ, which corresponds to a mass loss of 5 x 10⁻¹⁰ kg. The reactants weigh 0.5 micrograms more than the products, well below any scale's detection.
• Antimatter is not "reverse mass". An antiproton has positive mass identical to a proton. When matter meets antimatter, both masses convert to energy photons - it is the only process with 100% conversion efficiency.
• Nuclear weapons do not "split the atom into pure energy". The uranium atoms split into smaller atoms (barium, krypton, etc.) plus neutrons. Only the mass difference - about 0.09% - is released as kinetic energy and gamma rays.
• Photons are massless but carry energy. This is consistent with E² = (pc)² + (mc²)² giving E = pc when m = 0. A 500 nm green photon carries about 2.5 eV of energy with zero rest mass.

Energy Unit Reference

Unit	Value in Joules	Typical Use
Electron volt (eV)	1.602 x 10⁻¹⁹	Atomic and particle physics
Kilojoule (kJ)	1,000	Chemistry, food energy
Kilowatt-hour (kWh)	3,600,000	Electricity billing
Kilocalorie (kcal)	4,184	Nutrition (food Calories)
Ton of TNT	4.184 x 10⁹	Explosive yield comparison
British thermal unit (BTU)	1,055	Heating, HVAC
Megaton of TNT	4.184 x 10¹⁵	Nuclear yield comparison
Therm	1.055 x 10⁸	UK gas billing

Why c² Is Such a Large Number

The speed of light squared is 8.9875 x 10¹⁶ m²/s², meaning one kilogram of mass contains roughly 90 petajoules of energy. This is the reason nuclear reactions are so much more energetic than chemical ones: in chemistry, the binding energies involved are of order eV per atom (breaking electron bonds), while in nuclear reactions they are of order MeV per nucleon (breaking nuclear bonds) - a factor of about a million higher. The rest energy of a single gram of matter is equivalent to the output of a typical 1 GW nuclear power plant running for about 25 hours. It is also why fusion is theoretically so attractive: 1 kg of deuterium-tritium fuel, fused to completion, would release about 340 TJ, or enough energy to supply a UK household at average consumption (around 11 MWh per year per Ofgem 2025 figures) for over 8,500 years.

Binding Energy and the Mass Defect

The mass defect is the difference between the sum of individual nucleon masses and the actual mass of the bound nucleus. For a helium-4 nucleus, two protons and two neutrons weigh about 4.0320 amu separately, but the bound He-4 nucleus weighs only 4.0026 amu. The 0.0294 amu deficit - about 0.73% of the total - has been carried away as binding energy (about 28.3 MeV) when the nucleus formed. The binding energy per nucleon peaks near iron-56 (about 8.8 MeV per nucleon), which is why fusion of light elements releases energy (climbing toward the peak) while fission of heavy elements also releases energy (falling toward the peak from the heavy side). Any element past iron cannot release energy through either process - the iron peak marks the end of stellar nucleosynthesis in ordinary stars and is why supernovas are required to produce heavier elements.

Common Mistakes When Using E = mc²

• Mixing up units. Mass must be in kilograms and c in metres per second to get energy in joules. Using grams or c in km/s silently produces a wrong answer by orders of magnitude.
• Forgetting to square c. The equation is E = m times c-squared, not E = mc. Writing E = mc gives a result about 300 million times too small.
• Treating mass-energy as extractable. E = mc² gives the theoretical maximum, but no practical reaction on Earth extracts all of it. Even antimatter annihilation, which is 100% efficient in principle, is limited in practice by antimatter production costs - CERN's 2025 antiproton yield is measured in nanograms per year.
• Confusing relativistic and rest mass. Modern physics uses invariant rest mass only. "Relativistic mass" (m times γ) is an older convention discouraged since the 1990s; the mass in E = mc² is always the invariant rest mass.

For kinetic energy of everyday moving objects, the kinetic energy calculator uses KE = 1/2 mv². For photon energy from wavelength and frequency, the wavelength calculator applies E = hf = hc/λ. For nuclear decay problems, the half-life calculator models exponential decay of radioactive isotopes. All calculations run in your browser with no data sent anywhere.

Sources

• NIST - Speed of Light in Vacuum (c)
• NIST - Atomic Mass Constant (amu)
• NIST - Proton Mass Energy Equivalent (MeV)
• Rainville et al., Nature (2005) - A Direct Test of E = mc²
• CERN - The Higgs Boson
• IAEA - Nuclear Fusion Basics
• Einstein (1905) - Does the Inertia of a Body Depend Upon Its Energy Content?