   SEARCH HOME Math Central Quandaries & Queries  Question from Jim, a parent: Please explain the following formula: E=MC^2 (squared) Jim,

The formula gives the conversion factor between mass and energy, where c is the speed of light. When I was a kid I wondered which system of units it worked in - is there something special about the metric system? I asked the speaker at a public lecture, and he told me it is true in any consistent system of units, because energy has the dimensions mass * length2/time2 = mass * velocity2. [Compare the kinetic energy formula of Newtonian physics, E = 1/2 m v2.] It does NOT work if energy is measured in arbitrary units like BTU's or horsepower!

All energy has a mass associated with it. Thus, for instance, a wound-up clock weighs very slightly more than an unwound one (the difference is far too small to measure!) and a region of space with light travelling through it attracts bodies gravitationally in the same way as it would if it had a very small amount of matter in it instead. Very small: c is about 3 x 108 m/sec [it takes light about 1/8 second to go around the Earth; the delay is just noticeable on phone conversations carried by geostationary satellites] and its square, which you divide by to get the equivalent mass, is enormous. So one kilowatt-hour of energy (3.6 megajoules) would have a mass of about about 40 millionths of a microgram.

You may have heard that photons are "massless"; this means that they do not have a "rest mass" that remains even when they are stopped. A photon cannot be stopped; it can be slowed while interacting with matter, but cannot actually come to rest without ceasing to exist and having its energy absorbed by the matter that stopped it, rather like an ocean wave. When the photon leaves the matter that was slowing it it resumes its usual speed.

When neutrons and protons bond into an atomic nucleus, the resulting nucleus is more stable than the individual particles (otherwise it would not form!) and as a result it actually weighs less than its constituent parts. The difference in mass corresponds to the potential energy difference between the loose particles and the nucleus.

Conversely, under some circumstances (such as nuclear reactions) matter can be turned into energy. In many cases (eg, uranium fission) no subatomic particles are destroyed. The energy comes from redistributing the particles of a nucleus into more stable configurations. These more stable configurations have less mass than the original, and the remainder is converted to energy; most of the mass remains as matter. In other cases (eg, electron-positron annihilation) particles may actually be destroyed and their internal mass also converted to energy.
Because c is so big, mass-to-energy conversions tend to produce a huge amount of energy, even (as in atomic reactions) when only a small proportion of the mass is converted.

NB: Please do not try this at home!

RD     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.