E=mc² Calculator
Calculate the energy equivalent of a given mass using Einstein's famous equation E = mc², where c is the speed of light. Demonstrates the enormous energy contained in even small amounts of matter.
E = mc² is the most famous equation in physics. Derived by Albert Einstein in 1905 as a consequence of special relativity, it states that mass and energy are interchangeable forms of the same underlying quantity. The coefficient is the speed of light squared — about 9 × 10¹⁶ m²/s² — meaning even a tiny mass carries an astonishing amount of energy.
This single relationship explains where the Sun's energy comes from (mass loss in fusion), how nuclear power works (fission releases mass-energy), the principle behind the atomic bomb, and the origin of mass in particle physics. It also reveals that all mass — your body, this calculator, every atom — is essentially "frozen energy" in incredibly concentrated form.
Practical context: 1 kilogram of any matter, if fully converted to energy, releases ~9 × 10¹⁶ joules — equivalent to 21.5 megatons of TNT, roughly 1,500 Hiroshima-scale weapons. Of course, full conversion is extraordinarily difficult — matter-antimatter annihilation comes closest, but in practice we extract only fractions of a percent of mass-energy in nuclear fission (~0.1%) or fusion (~0.7%).
Common applications: nuclear engineering, particle physics, cosmology, astrophysics (stellar energy production), and the conceptual basis for understanding why mass exists. The calculator gives the equivalent energy for any input mass.
Inputs
Results
Energy
8.988e+13 J
Energy (kWh)
2.497e+7 kWh
TNT Equiv.
21.5 kt
E=mc² Results
| Parameter | Value |
|---|---|
| Mass | 0.001 kg (1.000000 g) |
| Speed of Light (c) | 299,792,458 m/s |
| c² | 8.987552e+16 m²/s² |
| Energy (J) | 8.987552e+13 J |
| Energy (kWh) | 2.4965e+7 kWh |
| Energy (GWh) | 2.4965e+1 GWh |
| TNT Equivalent (kt) | 21.48 kilotons |
| TNT Equivalent (Mt) | 0.0215 megatons |
| Formula | E = mc² |
Formula
How to use this calculator
- Enter the mass in kilograms (e.g., 0.001 for 1 gram).
- For very small masses, use scientific notation or μg/ng.
- Calculator returns the equivalent energy in joules.
- Compare to familiar references (TNT, kWh) for context.
- Real-world energy release is only a small fraction of E = mc² (nuclear ~0.1-0.7%).
- Antimatter annihilation is the only known full-conversion process.
Worked examples
Hiroshima bomb mass deficit
**Scenario:** Hiroshima bomb released ~15 kilotons TNT energy. How much mass was converted? **Calculation:** E = 15,000 × 4.184 × 10⁹ J = 6.28 × 10¹³ J. m = E/c² = 6.28 × 10¹³ / 9 × 10¹⁶ = 7 × 10⁻⁴ kg = 0.7 g. **Result:** Less than one gram of matter became energy. Only ~0.7% of the ~60 kg of uranium fuel actually fissioned, and only ~0.1% of that mass converted — so ~60 g of mass "left" the bomb in pure energy form.
Sun's energy output
**Scenario:** The Sun radiates 3.85 × 10²⁶ W. Mass loss rate? **Calculation:** Per second, ΔE = 3.85 × 10²⁶ J. Δm = ΔE / c² = 3.85 × 10²⁶ / 9 × 10¹⁶ = 4.28 × 10⁹ kg. **Result:** Sun loses ~4.3 million tonnes of mass *every second* as energy radiates away. Over 4.6 billion years: ~6 × 10²⁶ kg lost — about 0.03% of the Sun's mass. The proton-proton fusion chain (4H → He) accounts for this.
Bag of sugar energy content
**Scenario:** A 1 kg bag of sugar. Energy if mass fully converted? **Calculation:** E = 1 × 9 × 10¹⁶ = 9 × 10¹⁶ J. **Result:** ~9 × 10¹⁶ J — about 21.5 megatons TNT, roughly the largest hydrogen bomb ever built (Tsar Bomba ~50 Mt). For comparison, eating the sugar yields ~16 MJ chemical energy — only 0.0000000002% of its mass-energy. Chemistry barely touches the mass-energy reservoir.
When to use this calculator
**Use E = mc² for:**
- **Nuclear engineering**: fission, fusion, mass defect calculations. - **Particle physics**: rest energy, threshold energies for particle creation. - **Astrophysics**: stellar energy production, mass loss. - **Cosmology**: energy budget of the universe. - **Conceptual understanding**: mass and energy as a single quantity.
**Why "full conversion" is rare:**
The equation gives the total energy *equivalent* of mass. Most physical processes convert only a tiny fraction:
- **Chemical** (combustion, batteries): ~10⁻¹⁰ to 10⁻⁸ of mass. - **Nuclear fission** (U-235): ~10⁻³ of mass. - **Nuclear fusion** (H → He): ~7 × 10⁻³ of mass. - **Matter-antimatter**: 1.0 (full conversion). - **Black hole accretion**: up to ~0.4 (most efficient natural).
**Particle physics applications:**
- **Threshold energies**: creating new particles requires energy ≥ rest mass. - **Decay products**: total decay energy = Δmc² (Q-value). - **Collider design**: LHC delivers 13 TeV total energy → can create particles up to ~13 TeV mass.
**Cosmology:**
- **Total energy of universe**: includes dark matter (~27%), dark energy (~68%), ordinary matter (~5%). - **Big Bang nucleosynthesis**: 3 minutes of fusion turned ~25% of hydrogen into helium-4. - **Heat death scenarios**: depend on long-term mass-energy distribution.
**Mass defect example: deuterium**
Mass of proton + neutron: 938.272 + 939.565 = 1,877.837 MeV/c². Mass of deuterium (²H): 1,875.613 MeV/c². Defect: 2.224 MeV = binding energy.
Form one deuterium → release 2.224 MeV. Fusion of D + T → ⁴He + n releases 17.6 MeV.
**Practical nuclear energy:**
- **Fission reactor**: 200 MeV per U-235 fission. 1 kg U-235 → ~24,000 MWh thermal energy. - **Fusion (ITER, future)**: 17.6 MeV per D-T fusion. 1 kg D-T mix → ~340,000 MWh.
For comparison: 1 kg gasoline → ~12 kWh combustion. Nuclear is ~10⁶ times more energy-dense.
**Antimatter as fuel (theoretical):**
- 1 kg antimatter + 1 kg matter = 1.8 × 10¹⁷ J ≈ 43 megatons TNT. - Annihilation produces gamma rays (511 keV for e⁺/e⁻) or pions (proton-antiproton). - Storage requires magnetic trapping in vacuum. - Production: extremely inefficient — most expensive substance ever (~$100 trillion/g).
**Conservation laws:**
- **Mass + energy** is conserved (not mass alone or energy alone separately for nuclear/particle physics). - **Mass-energy** + **momentum**: conserved as a 4-vector. - **Charge, baryon number, lepton number**: separately conserved.
**Software:**
- **NIST Atomic Mass Tables**: precise masses for binding energy calculations. - **MCNP, Geant4**: nuclear reaction simulation. - **PYTHIA, MadGraph**: particle physics event generation.
**Pitfalls:**
- **Confusing rest mass with relativistic mass**: modern convention is invariant rest mass m₀. - **Mixing units**: SI vs natural units (eV, MeV, GeV). - **Assuming free energy**: real conversion is far from 100% efficient. - **Sign conventions**: kinetic energy = total − rest energy. - **Forgetting c is huge**: factor ~10¹⁶ amplifies tiny masses into huge energies.
Common mistakes to avoid
- Assuming mass can be freely converted to energy at 100% efficiency in practice.
- Confusing rest mass and relativistic mass (modern usage: rest mass is the "mass").
- Forgetting to square c (factor of 3 × 10⁸ → 9 × 10¹⁶).
- Mixing units (J vs eV vs kWh vs TNT-equivalent).
- Treating chemical energy releases as comparable to nuclear (off by ~6 orders of magnitude).
- Confusing binding energy with rest mass energy.
- Forgetting conservation of charge, baryon number, lepton number alongside mass-energy.
- Applying classical E = ½mv² where relativistic E = γmc² is needed.