Sculpture of Einstein’s equation at the
2006 Walk of Ideas, Berlin, Germany.

This would be proven experimentally by Cockcroft and Walton in 1932, when they developed the first particle accelerator and used it to smash protons into a lithium target to produce alpha particles. They had split the atom. Six years later nuclear fission would be discovered.
The original Cockcroft and Walton experiment. They bombarded a lithium target with protons, splitting the atom to create two alpha particles and converting some of its mass to energy. 
Both Einstein and Szilárd would come to regret that letter. They were against using the atom bomb against civilians and thought that simply demonstrating the power of the bomb would be enough to force Japan’s surrender. After the war, Szilárd was so horrified by atomic weapons that he gave up physics for molecular biology.
The EinsteinSzilárd letter that was sent to President Roosevelt. 
Incidentally, E = mc^{2} is just part of Einstein’s equation, and only describes the energy of an object at rest. The full equation is E^{2} = (mc^{2})^{2} + (pc)^{2}, where p is the momentum. If the object is at rest then it has zero momentum and the equation reduces to E = mc^{2}. If the object is a massless particle like a photon, then the equation reduces to E = pc, which means that the energy of a massless particle is equal to its momentum multiplied by the velocity of light.
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