DERIVATION OF E = MC2
I. FORCE OF AN ABSORBED LIGHT BEAM
Light hits matter and the electric field shakes the heck out of the electrons. Then the movement of the electrons in the magnetic field of the light wave causes the force on the body. E will be used for electric field until we get to II below.
Consider a light beam going in the x direction, polarized in the y direction, hitting a surface in the yz plane. The power absorbed by a charge q is Fvy , where F is Eq, electric field times charge, and vy is the velocity component. The magnetic force on q is f = qvxB. Recall that the Poynting vector is S=(1/m o )ExB, and show that f is in the same direction as S, the x direction in this case. Therefore the magnitude of the force is f = qvy B. Eliminate vy from these equations and use B = E/c in a light wave, and show that the force f = power absorbed/c.
Note also that the impulse (force×time) is energy absorbed/c.
Hence when light strikes a free object, the momentum given it is energy
absorbed/c. In the following, E will be used for energy, because who ever heard
of
U = mc2 ?
II. CENTER OF MASS
Consider a box floating freely in space with no forces acting on it. Inside the box there is a projectile launcher at one end that is set to go off at a certain time. It will send a mass to the other end, where it will hit and stick. When the thing is launched, the box receives a kick and starts to move. Then it stops when it receives the opposite impulse at the other end. You know that with no external force acting, the total momentum is constant (zero in this case), and that an equivalent statement is that the center of mass has constant velocity (zero in this case). This is a fundamental law of nature: Conservation of Momentum.
We could imagine a possible violation of this law if we replace the projectile with a light pulser. When a pulse of light is emitted, the box will move in the opposite direction until the light is absorbed at the other end. In the projectile case, the movement was compensated by the projectile's mass, so the center of mass of the system does not move. Now of course a light pulse delivers so little impulse and is in flight for so little time that it is fair to ask why we should bother with this problem at all. Be patient; it will pay off, it’s a gedanken experiment of Einstein’s (thought experiment).
III. MASS-ENERGY EQUIVALENCE
Suppose our law is obeyed exactly. Then the light pulse must have delivered mass to the other end. Let's find out how much. To simplify, let our apparatus be the equivalent of mass M on each end separated by distance 2X. Let the light pulse have energy E. When it is flashed, the impulse to the flasher is E/c, so the apparatus moves with speed E/2Mc for a time 2X/c. Then the pulse is absorbed at the other end, bringing the apparatus to a stop. The apparatus has moved a distance x = vt = (E/2Mc)(2X/c) = EX/Mc2 . In order to have the center of mass not move, assume mass m has been transferred from flasher to target, and solve for m:
(M-m)(X+x) = (M+m)(X-x)
Solving for m, we find that m = Mx/X which reduces to E/c2 . There is overwhelming evidence that this is always true: if you transfer energy, you have transferred mass equal to the energy/c2 . Compressing a spring adds a tiny amount of mass to the spring. (This is not detectable unless it is a nuclear spring.) Back to main physics page, or… More discussion of E = mc2 .
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fluids, heat, electricity
and magnetism
vibrations and waves
quantum
index with short explanations
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