E = mc² Calculator

Background.

The equation E = mc² is the most recognizable formula in physics, expressing the equivalence of mass and energy. It states that the rest energy E of a body is equal to its mass m multiplied by the square of the speed of light in vacuum c. This relationship, derived by Albert Einstein in 1905 as a consequence of special relativity, revolutionized physics by demonstrating that mass is not conserved independently of energy and that even a small amount of mass corresponds to an enormous amount of energy due to the c² factor. An E = mc² calculator performs the numerical conversion between mass and energy, expressing the result in joules or in explosive yield equivalents such as kilotons or megatons of TNT.

The practical implications of mass-energy equivalence became apparent with the discovery of nuclear fission in 1938 by Otto Hahn and Fritz Strassmann, and its subsequent interpretation by Lise Meitner and Otto Frisch. In fission, a heavy nucleus such as uranium-235 splits into lighter fragments whose combined mass is slightly less than the original nucleus. This mass difference, called the mass defect, is converted into kinetic energy of the fragments and released gamma radiation according to E = mc². The energy released per fission event is approximately 200 MeV, which corresponds to a mass defect of about 0.2 atomic mass units, or 3.2 × 10⁻²⁸ kilograms. While this mass is imperceptibly small, the enormous value of c² means that one kilogram of uranium-235 undergoing complete fission releases energy equivalent to the detonation of roughly 17,000 metric tons of TNT.

Nuclear fusion in stars operates on the same principle. In the Sun's core, four protons combine through the proton-proton chain to form a helium-4 nucleus. The mass of the helium nucleus is about 0.7 percent less than the mass of the four protons. This mass defect, multiplied by c², yields the 3.8 × 10²⁶ watts of power radiated by the Sun. Every second, the Sun converts approximately 4.3 million metric tons of mass into energy. On Earth, thermonuclear weapons and experimental fusion reactors such as ITER aim to harness this process. A fusion-based power plant would require only a few grams of deuterium and tritium fuel to produce the same energy as many tons of coal, with helium as the only direct byproduct.

Outside nuclear physics, E = mc² governs particle-antiparticle annihilation, the operation of particle accelerators, and the energetics of chemical reactions. In positron-electron annihilation, the entire rest mass of both particles converts into two 511-keV gamma photons. In high-energy physics, the Large Hadron Collider accelerates protons to 6.5 TeV, at which speed their kinetic energy is thousands of times their rest energy. Even chemical reactions involve minute mass changes, though these are far below measurement precision for ordinary laboratory balances. The combustion of one kilogram of gasoline releases approximately 44 MJ, corresponding to a mass loss of 4.9 × 10⁻¹⁰ kilograms, which is undetectable without mass spectrometry. The calculator therefore serves primarily nuclear, astrophysical, and educational contexts where the mass-energy conversion is substantial.

What is e = mc² calculator?

Mass-energy equivalence is the principle that mass and energy are two manifestations of the same physical quantity, related by the equation E = mc². In this equation, E is the rest energy of a body, m is its rest mass, and c is the speed of light in vacuum, exactly 299,792,458 meters per second. The equation is not a statement that mass can be converted into energy or vice versa in the sense of transmutation; rather, it asserts that any change in the energy of an isolated system is accompanied by a proportional change in its mass, and any change in mass is accompanied by a proportional change in energy.

The factor c² has the numerical value of approximately 8.98755 × 10¹⁶ joules per kilogram. This means that one kilogram of matter, if entirely converted to energy, would yield roughly 90 petajoules. For comparison, the total annual electricity generation of a large country is on the order of hundreds of petajoules. Because c is so large, the mass equivalent of everyday energies is extremely small. A 100-watt light bulb operating for one year consumes about 3.15 × 10⁹ joules, corresponding to a mass equivalent of 3.5 × 10⁻⁸ kilograms, or 35 micrograms. The equation is exact within special relativity and has been verified experimentally to high precision in nuclear reactions and particle physics.

How to use this calculator.

1. 01Select the calculation mode: compute energy from mass, or compute mass from energy.
2. 02Enter the numerical value: mass in kilograms, or energy in joules.
3. 03Select the output unit: joules, kilotons of TNT, or megatons of TNT.
4. 04Click Calculate to obtain the converted value.
5. 05Review the result and note that the speed of light used is the exact SI-defined value.

The formula.

Einstein derived E = mc² in his 1905 paper "Does the Inertia of a Body Depend Upon Its Energy Content?" by considering the emission of light from a body and applying the Lorentz transformations of special relativity. The derivation begins with the relativistic expression for the kinetic energy of a particle, K = (γ − 1)mc², where γ = 1 / √(1 − v²/c²) is the Lorentz factor. Expanding this in a Taylor series for v << c yields K ≈ ½mv², the classical kinetic energy, plus higher-order terms. The term mc² that remains even when v = 0 is interpreted as the rest energy of the particle, independent of its motion.

The total energy of a free particle is E = γmc², which reduces to E = mc² when the particle is at rest in the chosen inertial frame. For a system of particles, the total energy is the sum of rest energies and kinetic energies. If the system is bound, as in an atomic nucleus, the binding energy reduces the total mass below the sum of the constituent masses. This mass defect is precisely the binding energy divided by c². For helium-4, the binding energy is approximately 28 MeV, so the nuclear mass is 28 MeV/c² less than the mass of two protons and two neutrons. The calculator applies the rest-energy formula E = mc² directly; for kinetic energies at relativistic speeds, the user must compute γmc² separately.

TNT equivalent is a conventional unit of energy defined by the energy released in the detonation of trinitrotoluene. The standard definition, adopted by NIST and NATO, is 1 kiloton TNT = 4.184 × 10¹² joules, which is exactly 10¹² calories in the thermochemical definition. One megaton TNT is 10³ kilotons, or 4.184 × 10¹⁵ joules. These definitions are arbitrary but internationally standardized, allowing comparison of nuclear yields with conventional explosive quantities. The calculator converts joules to TNT equivalents by simple division; the user should be aware that actual explosive effects depend on blast geometry, altitude, and energy partitioning, not just total energy.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Einstein, A. (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" Annalen der Physik 18:639–641. doi:10.1002/andp.19053231314
2. [2]BIPM (2019). The International System of Units (SI Brochure), 9th ed.
3. [3]Serway, R.A. and Jewett, J.W. (2018). Physics for Scientists and Engineers, 10th ed. Cengage. ISBN 978-1337553278.
4. [4]CODATA (2018). Recommended values of physical constants. NIST.
5. [5]Cohen, E.R. and Taylor, B.N. (1987). "The 1986 Adjustment of the Fundamental Physical Constants." Rev. Mod. Phys. 59:1121. doi:10.1103/RevModPhys.59.1121