An interesting fact about light is that the path it takes from A to B always min

Question

An interesting fact about light is that the path it takes from A to B always minimizes the total light travel time. In other words, any other path from A to B, including the shortest-distance path, would take the light more time. One can show that this is equivalent to Snell's law. In the case shown in the picture, the time it takes the light to travel from point A to the green point, on the interface, is 31.0 ns. By using Snell's law and considering the geometry, answer the following.

(a) What is the x-coordinate of the green point? (in meters)


(b) What is the index of refraction of the second medium?


(c) What is the total time, in nanoseconds, it takes the light to travel from point A to point B?

Explanation / Answer

We can find the distance between A and the green spot using d = vt

AO = vt = 3*10^8m/s * 31ns = 6.3 m

Now this is the hypotenuse of the upper triangle. For 1 we thus have:

cos1 = 5/6.3
=> sin1 = sqrt(1-cos^2 1)
= 0.608

The horizontal position of the green dot O is therefore

=> Xo = 6.3 m * sin1
= 3.83 m

b)
To figure out the index we need to use Snell’s law and for that we need the sine of the angles
which is the ratio of opposite over hypotenuse and so:

n2/n1 = sin1/sin2

= (xo/Ao)/(10-xo/OB)

We first need to find OB, which is the hypotenuse of the lower triangle:

OB = sqrt(5^2 + (10-xo)^2)
= 7.941m

n2 = (3.83/6.3)/((10-3.83)/7.941)
= .7824

c)
t2 = OB/v2
= OB/c/n2

= 7.941 * 0.7824 / 3*10^8
= 20.7 ns

So, total time = 31+20.7 = 51.7ns

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