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Do photons have mass? Adam (student)
The Doctor Responds: Okay, a serious question (succinct and to the point) requires an equally serious answer. Your question raises some nontrivial issues. Einstein defined the photon as a massless particle of pure electromagnetic energy. Unfortunately, Einstein also described a massenergy equivalency. So, if photons are energetic, they must have mass. How we do resolve Einstein's contradiction? By postulating the concept of relativistic mass. Try this experiment: grab hold of a photon (this may be more difficult than it sounds, because photons are both remarkably quick and uncommonly slippery), place it on a triple beam balance, and measure its mass. As long as it remains stationary on the scale, the photon is indeed massless (just as Einstein advertised). So far, so good. Except that photons don't do a very good job remaining motionless. In fact, they tend to propagate at a fixed velocity (which we call the speed of light). In free space, that velocity is 300 million meters per second. (In media other than vacuum, photons move more slowly; your mileage may vary.) The speed of light is, of course, a relativistic velocity  which is how photons in motion acquire relativistic mass. Planck's Law tells us that the energy of a photon is frequency dependent, and can be quantified as
where e_{p} represents the energy (in Joules) of a single photon, nu is the photon's frequency, in cycles per second, or Hz, and h, Planck's Constant, is a fudge factor to make all the units come out with dimensional consistency. (The accepted value for Planck's Constant is 6.626*10^{34} J*s; glad you asked.) Relativity theory tells us that the energy of a particle is related to its mass and the speed of light by the familiar relationship:
This suggests that if we know the mass of a photon, since we know the value of its velocity (c), we can compute its energy. Conversely, if we know the energy of a photon, we should be able to rearrange relativity, and solve for its equivalent mass. The next step is to assume the Planck energy of a photon and its relativistic energy are equal. This would leave us with:
[Planck = Einstein] and a little algebraic manipulation yields an expression for the (frequencydependent) relativistic mass of a photon:
Let's try this relationship on for size with a familiar photon emitted by the spinflip of neutral hydrogen. The resulting frequency, 1420 MHz, is a popular spot on the SETI dial. Plugging in this frequency, Planck's constant, and the speed of light, we see that an H1 photon has a relativistic mass on the order of 1*10^{41} kg. Not terribly heavy, to be sure, but still, clearly, a nonzero mass. And hydrogen line photons aren't even Catholic.

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