Physics · Particle & Nuclear Physics · Relativistic Physics
Mass-Energy Equivalence Calculator
Calculates the rest energy of a mass using Einstein's mass-energy equivalence formula E = mc².
Calculator
Formula
E is the rest energy in joules (J), m is the rest mass in kilograms (kg), and c is the speed of light in a vacuum (2.998 \times 10^8 \text{ m/s}). The formula shows that mass and energy are interchangeable, with c² acting as the conversion factor between the two quantities.
Source: Einstein, A. (1905). 'Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?' Annalen der Physik, 18, 639–641. (On the Electrodynamics of Moving Bodies)
How it works
Einstein's 1905 special theory of relativity introduced the radical notion that mass is a concentrated form of energy. The rest energy of an object is the energy it possesses solely by virtue of having mass, independent of any motion or position. Even a tiny mass corresponds to an enormous amount of energy because the conversion factor, c² ≈ 8.99 × 10¹⁶ m²/s², is astronomically large. This is why nuclear reactions — which convert only a tiny fraction of their fuel mass into energy — can release devastatingly large amounts of power.
The formula E = mc² applies specifically to an object at rest. Here, E is the rest energy in joules (J), m is the rest mass in kilograms (kg), and c = 2.998 × 10⁸ m/s is the speed of light in vacuum. For a moving object, the full relativistic energy expression E² = (pc)² + (mc²)² must be used, where p is the relativistic momentum. The mass-energy equivalence formula is the zero-momentum special case of this more general relation. This calculator supports mass inputs in kilograms, grams, milligrams, atomic mass units (u), and pounds, converting internally to SI before applying the formula.
Practical applications span nuclear and particle physics, cosmology, and engineering. In nuclear reactors, the small mass deficit between reactants and products (the binding energy difference) is converted to heat according to E = mc². In particle accelerators like CERN's LHC, rest-mass energies expressed in MeV or GeV are the fundamental currency for describing collisions and decay processes. In astrophysics, the enormous luminosity of stars is explained by the continuous mass-to-energy conversion occurring through fusion reactions in stellar cores. The TNT equivalent output provides intuitive context by comparing the rest energy to the chemical energy of conventional explosives.
Worked example
Consider calculating the rest energy of a single proton, with a rest mass of 1.6726 × 10⁻²⁷ kg.
Step 1 — Identify the mass: m = 1.6726 × 10⁻²⁷ kg (the proton rest mass). If entering in atomic mass units, 1 u = 1.66054 × 10⁻²⁷ kg and a proton is approximately 1.00728 u.
Step 2 — Apply E = mc²:
E = (1.6726 × 10⁻²⁷ kg) × (2.998 × 10⁸ m/s)²
E = (1.6726 × 10⁻²⁷) × (8.988 × 10¹⁶)
E ≈ 1.5033 × 10⁻¹⁰ J
Step 3 — Convert to MeV: Dividing by 1.602 × 10⁻¹³ J/MeV gives approximately 938.3 MeV. This is the well-known proton rest mass energy, a fundamental constant in particle physics.
Step 4 — TNT equivalent: Dividing by 4.184 × 10¹² J/kt gives approximately 3.59 × 10⁻²³ kt, an infinitesimally small but meaningful reference for comparing particle-scale and macroscopic energies.
For a second example, consider 1 gram (0.001 kg) of any matter: E = 0.001 × 8.988 × 10¹⁶ ≈ 8.99 × 10¹³ J, equivalent to roughly 21.5 kilotons of TNT — comparable to the Hiroshima atomic bomb yield, illustrating how small mass corresponds to enormous energy.
Limitations & notes
This calculator computes rest energy only, applicable to objects with zero velocity in the reference frame of interest. For moving particles, the full relativistic energy-momentum relation E² = (pc)² + (mc²)² must be used, and kinetic energy must be computed as the difference between total relativistic energy and rest energy. The formula also assumes classical matter with a well-defined rest mass; massless particles (photons, gluons) have zero rest mass and carry energy entirely through momentum (E = pc). At nuclear scales, the effective mass involved in a reaction is the mass deficit between initial and final states — not the total mass of the nuclei — so users must calculate the mass difference carefully before applying this calculator. The formula does not account for binding energy corrections within composite particles at high precision. All outputs assume the exact CODATA value c = 299,792,458 m/s.
Frequently asked questions
What does E = mc² actually mean?
E = mc² states that mass and energy are equivalent and interchangeable. An object at rest with mass m possesses a rest energy E equal to that mass multiplied by the square of the speed of light. This means even a tiny amount of mass contains an enormous amount of stored energy, because c² ≈ 9 × 10¹⁶ m²/s² is a very large number.
How is the mass-energy equivalence used in nuclear reactions?
In nuclear fission and fusion, the total mass of the products is slightly less than the total mass of the reactants. This mass deficit, called the mass defect, is converted into energy according to E = mc². For example, in uranium fission, approximately 0.1% of the nuclear mass is converted to energy, producing the enormous heat that drives nuclear power plants and weapons.
Why is the proton rest mass energy expressed as 938.3 MeV?
In particle physics, it is conventional to express rest energies in megaelectronvolts (MeV) rather than joules, because the energies involved in particle interactions are on this scale. One MeV = 1.602 × 10⁻¹³ J. Applying E = mc² to the proton mass of 1.6726 × 10⁻²⁷ kg and converting gives exactly 938.272 MeV, a fundamental constant used throughout particle and nuclear physics.
Does E = mc² apply to massless particles like photons?
No. Photons have zero rest mass, so E = mc² gives zero, which is incorrect. For massless particles, the correct relation is E = pc, where p is the photon's momentum. The full relativistic energy-momentum relation E² = (pc)² + (mc²)² reduces to E = pc when m = 0, correctly describing photons and other massless particles.
What is the speed of light value used in this calculator?
This calculator uses the exact defined value c = 299,792,458 m/s, giving c² = 8.987551787368176 × 10¹⁶ m²/s². This value has been exact by definition since 1983, when the metre was redefined in terms of the speed of light. There is therefore no uncertainty in c itself; measurement uncertainty enters only through the mass input.
Last updated: 2025-01-15 · Formula verified against primary sources.