E = mc² Calculator – Mass–Energy Equivalence

What does E = mc² mean?

Einstein’s mass–energy equivalence formula \[ E = mc^2 \] states that mass \(m\) and energy \(E\) are two forms of the same physical quantity. A small amount of mass corresponds to a huge amount of energy because the speed of light \(c\) is very large and it is squared.

The calculator on this page lets you work in both directions:

• Mass → Energy: given a mass, compute the equivalent energy.
• Energy → Mass: given an energy release, compute the mass defect.

How to calculate energy from mass (step by step)

1. Convert the mass to kilograms.
   Our tool does this automatically, but conceptually you should always work in SI units first.
2. Use \(E = mc^2\).
   Multiply the mass in kilograms by \(c^2\): \[ E[\text{J}] = m[\text{kg}] \times c^2 \]
3. Convert joules to convenient units.
   For example: \[ 1~\text{kWh} = 3.6 \times 10^6~\text{J} \] \[ 1~\text{eV} = 1.602176634 \times 10^{-19}~\text{J} \]

Example: 1 gram of matter

1 g = 0.001 kg.

\[ E = (0.001~\text{kg}) \times (2.99792458 \times 10^8~\text{m/s})^2 \approx 8.99 \times 10^{13}~\text{J} \]

In kilowatt-hours: \[ E \approx \frac{8.99 \times 10^{13}~\text{J}}{3.6 \times 10^6~\text{J/kWh}} \approx 2.5 \times 10^7~\text{kWh} \] That’s tens of millions of kWh from just one gram of matter.

How to calculate mass from energy

Rearranging \(E = mc^2\) gives:

1. Convert the energy to joules.
2. Divide by \(c^2\) to get mass in kilograms.
3. Convert kilograms to grams, atomic mass units, or tons if desired.

Example: 1 kiloton of TNT

By convention, 1 ton of TNT ≈ \(4.184 \times 10^9\) J, so 1 kiloton (kt) ≈ \(4.184 \times 10^{12}\) J.

\[ m = \frac{4.184 \times 10^{12}~\text{J}}{8.98755179 \times 10^{16}~\text{m}^2/\text{s}^2} \approx 4.65 \times 10^{-5}~\text{kg} \]

That’s only about 46.5 milligrams of mass converted to energy.

Common conversion factors for E = mc²

• 1 kg → \(8.98755179 \times 10^{16}\) J
• 1 g → \(8.98755179 \times 10^{13}\) J
• 1 u → ≈ 931.494 MeV
• 1 J → \(1 / c^2 \approx 1.11265 \times 10^{-17}\) kg

Where is E = mc² used in practice?

• Nuclear physics: computing binding energies and reaction Q-values.
• Particle physics: converting between particle masses (in MeV/c²) and energies.
• Astrophysics: estimating energy output from fusion in stars.
• Everyday relativity: in principle, any energy stored in a system slightly increases its mass.

Limitations and misconceptions

• Rest energy vs total energy: \(E = mc^2\) usually refers to rest energy. For moving objects, the full relativistic energy is \[ E^2 = (pc)^2 + (m c^2)^2 \] where \(p\) is momentum.
• “Relativistic mass” is outdated: modern treatments keep mass constant and let energy and momentum change with speed.
• Not all mass is easily converted: in real reactors or bombs only a fraction of the available mass is converted to usable energy.

FAQ

What does E = mc² actually mean?

It means that mass and energy are interchangeable. A system with more energy has more mass, even if that extra mass is far too small to measure with ordinary scales.

Which value of c does this calculator use?

It uses the exact CODATA value \(c = 299{,}792{,}458~\text{m/s}\), so results are consistent with modern physics references.

Can I use this for nuclear binding energies?

Yes. Enter the mass defect (difference between initial and final mass) and choose MeV as the output unit. For convenience, remember that 1 u ≈ 931.494 MeV.

Why are the numbers so huge?

Because \(c^2\) is enormous. That’s why even tiny mass changes in nuclear reactions can release so much energy.