Why Does E=mc2_ - Brian Cox [58]
Imagine a bucket of armed mousetraps, all storing energy in the springs. We know that wound-up springs store energy because when the trap is triggered there is a loud bang (which is energy being released as sound) and the trap might jump up in the air (energy being turned into kinetic energy). Now imagine that one trap goes off and triggers the rest. There is a huge clatter as the energy stored in the springs is liberated and the mousetraps snap shut. The conservation of energy says that the energy before the mousetraps snap shut must equal the energy afterward. Moreover, since the traps were initially all sitting at rest, the total energy must equal mc2, where m is the total mass of the bucket of primed traps. Afterward, we have a bunch of spent traps plus the energy that was liberated. To balance the energy before with that afterward, it therefore follows that the bucket of armed mousetraps is actually more massive than the bucket of triggered traps. Let’s think of another example, this time involving a contribution to mass arising from kinetic energy. A box full of hot gas has more mass than an identical box containing the same gas at a lower temperature. The temperature measures how fast the molecules are whizzing around inside the box—the hotter the gas, the faster the molecules move around. Because they are moving faster, they have more kinetic energy (i.e., the result of adding together the values ofmv2 for each molecule is bigger for the hot gas) and hence the box has more mass. The logic extends to everything that stores energy. A new battery is more massive than a used battery, a hot flask of coffee is more massive than a cold one, and a steaming-hot meat and potato pie bought at halftime on a wet Saturday afternoon at Oldham Athletic’s football ground is more massive than the same uneaten pie at the end of the game.
The conversion of mass to energy is therefore not such an exotic process. It is happening all the time. As you relax by a crackling fire you are absorbing heat from the burning coals, and that heat takes energy away from the coal. In the morning, when the fire has died away, you could very carefully sweep up every last piece of ash and weigh it with scales of unfeasible accuracy. Even if you miraculously managed to get every atom of ash, you would find that it weighed less than the original coals weighed. The difference would be equal to the amount of energy liberated divided by the speed of light squared, as predicted by E = mc2, i.e., according to m = E/c2. We can quickly figure out how tiny the change in mass would be for the kind of fire that might warm your house as the night draws near. If the fire generates 1,000 watts of power for 8 hours, then the total energy output is equal to 1,000 x (8 x 60 x 60) joules (because we have to work in seconds, not hours, in order to get an answer in joules), which is just less than 30 million joules. The corresponding loss of mass must therefore be equal to 30 million joules divided by the speed of light squared, and that is equal to less than one-millionth of a gram. The explanation for the tiny reduction in mass is a direct consequence of the conservation of energy. Before igniting the fire, the total energy of the coals is equal to the total mass of coal multiplied by the speed of light squared. As the fire burns, energy leaves the fire. Eventually, the fire dies and we are left with ash. According to the law of conservation of energy, the total energy of the ash must be less than the total energy of the coal by an amount equal to the energy that went into warming the room.