# Refractive index question - light beam around the world

1. Jan 3, 2012

1. The refractive index of the Earth’s atmosphere is
n = 1.01 + α(R − r),
where α is a constant, r is the radial distance from the Earth’s centre and
R = 6.4 × 10^6 m is the Earth’s radius. By considering a path comprising a series of
total internal reﬂections or otherwise, ﬁnd a value of α for which a light ray emitted
horizontally close to the Earth’s surface would go around the Earth. (The eﬀects of
absorption may be ignored and the Earth may be taken to be a perfect sphere of radius
R.)

2. Relevant equations
n1sin(x1)=n2sin(x2)

3. The attempt at a solution
I really have no idea what to do as I understand the refractive index is constantly changing?
Don't know where to start, any help would be much appreciated.

2. Jan 3, 2012

### rude man

Think of a wavefront launched horizontally. Realize that the top of this wavefront will move faster than the bottom, since n is lower at higher altitudes ... yet the wavefront phasing across its front has to be the same for all heights - whar does that requirement impose on grad(n)?

3. Jan 4, 2012

Thank you very much for your reply, are you suggesting the light curves around the world? I considered this but thought that only really happened at black holes, or have I misunderstood your response?

4. Jan 4, 2012

### rude man

I never heard of the herm 'light curves around the world' per se but that is what I had in mind. And I know next to zilch about black holes.

Can you go from there?

BTW the problem did not state that it was possible, it just asked what the n gradient had to be were it possible.