Light

If there were no atmosphere, there would be no atoms to scatter any of the light from the sun downwards towards us and the sky would appear black (as it appears on the moon!).

The problem basically gives us that E _sâàù = , where V is the volume of the scatterer and K incorporates anything (including constants) that we might have left out. Clearly the quantity VK/r must be dimensionless, so K must have units of (length) ^-2 . The only other factor that the scattering might reasonably depend on is the wavelength. So K = f (λ) . λ has units of length so Kâàùλ ^-2 so E _i/E _sâàùλ ^-2 .

The time taken to travel through each layer is given by t _i = d _i/v _i so the total time is a sum:

But since v = c/n , 1/v _i = n _i/c so we can write the expression as:

Take two points P and Q above the surface. The light moves from P , reflects off the surface and ends up at Q . A normal to the surface and a line joining P and Q define a plane called the plane of incidence. Choose some point R on the surface, but also in the plane of incidence (these define a line L ); we will show a ray must reflect at some such point. Choose another point R' also on the surface, but not in the plane of incidence. For any such point it must be true that we can draw a line from R' to a point R perpendicular to L . The distance PR' is then given by which is by definition greater than PR , unless R' lies on L (we have defined it not to, though). Similarly the distance R'Q > RQ since R'Q = . Thus for any point on the surface but not on the plane of incidence we can find a shorter path of the reflecting ray through a point in the plane of incidence. Thus, by Fermat's principle, it should take this shortest path (since we are dealing with a single medium, shortest distance corresponds to shortest time). Since R , P and Q are all in the plane of incidence, PR and RQ must lie in the plane of incidence, which also contains the normal to the surface (we defined it this way).

Using the result of the last problem we will assume P the point from which the light comes, Q the point to which the light goes, and R , the point from which the light reflects are all coplanar. Once again, since we are considering a single medium only the path of least time will be the path of least distance. Let the x- axis correspond to the surface. Without loss of generality let both P and Q be a distance y ₀ above the surface, with P on the y -axis and Q a distance x ₀ away (at (x ₀, y ₀) ). The point R can be anywhere along the x -axis. Call the coordinate of R (x ₁, 0) . The distance PR is given by . The distance from RQ is given by . Thus the total distance PRQ is:

Now we want to minimize this distance with respect to x ₁ so we can take the derivative and set it equal to zero:

Canceling the radicals this yields x ₁ + x ₁ - x ₀ = 0 . Thus x ₁ = x ₀/2 . Thus the point of reflection must fall exactly half way between P and Q . It follows immediately from symmetry that the incident and reflective angles must be equal.