1. Stars twinkle. Could this be due to the statistical fluctuations of numbers of photons entering your eyes? If there are N photons per second, the fluctuation in number will be of the order of the square root of N. The twinkle of a star is a substantial fraction of its brightness. So you need to figure out how many photons per second enter your eyes from a typical star. Imagine yourself in a dark field at night, and somewhere in the distance you can see a 100 watt bulb. About how far away is the bulb if it’s about as bright as a star? If you assume the bulb puts out 5% of its energy as visible light, how many photons a second does it emit? How many of those enter your eyes? What can you conclude about the twinkling star?

2. Consider the red hot glowing element in a stovetop. Find out its power, estimate its surface area and temperature, and from this come up with an order of magnitude estimate of Stefan’s constant. At about what wavelength does it put out most power?

3. In a dielectric medium, the energy-momentum relationship for photons becomes

E = (c/n)p

where n is the index of refraction.

Show that for a relativistic charged particle moving through a dielectric medium, it is possible for the particle to emit a photon and thus lose energy. (You showed in an earlier assignment that this couldn’t happen for a particle moving through empty space – so why doesn’t the same argument work in this situation?)

Consider a very fast particle (say, a proton at 10 Gev) emitting a visible-light photon. Write down the equations for conservation of energy and momentum, and, by making suitable approximations, find the angle at which the photon is emitted. Make some reasonable assumption for n, say n = 1.5.

Now imagine that as the proton moves through the medium, as it passes each point a sphere of radiation ripples out from that point at the speed of light in the medium, that is, at c/n. By putting all these expanding spheres together at some instant in time, try to construct the wavefront of the emitted radiation. Can you relate this to the angle of emission of the photon found above?

(This radiation is called Cerenkov radiation, and is the basic mechanism of many particle detectors.)