Derivation of Snell’s Law of Refraction
Refraction at a boundary between two transparent surfaces
A wavefront moves from XY to X’Y’ in time t from medium 1
with refractive index n1 and speed c1 to medium 2 with
refractive index n2 and speed c2. The incident angle is i and
the angle of refraction is r.
Write down equations for the lengths XX’ and YY’ in terms
of the speed of the wave in the different media and the
time.
Medium 1
n1
c1
YY’
XX’
Medium 2
n2
c2
Write down equations for the lengths XX’ and YY’ in terms of the length XY’ and the angles i and r.
Now equate your two equations for XX’ and YY’ to obtain two equations which link the speed of the wave to the
length XY’, time and the angles i and r.
By putting both equations together, eliminate XY’ and time.
Rearrange your equation to obtain an expression for
sin 𝑖
sin 𝑟
in terms of the speed of light in the two different media.
Refractive index of medium 2 with respect to medium 1
1𝑛2
Therefore:
=
sin 𝑖
sin 𝑟
Absolute refractive index
The absolute refractive index is calculated for a material with respect to a vacuum (i.e. medium 1 is a vacuum). Thus
c1 = c, c2 = cs and 1n2 = ns.
𝑅𝑒𝑓𝑟𝑎𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑑𝑒𝑥 =
𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑙𝑖𝑔ℎ𝑡 𝑖𝑛 𝑎 𝑣𝑎𝑐𝑢𝑢𝑚
𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑙𝑖𝑔ℎ𝑡 𝑖𝑛 𝑎 𝑡𝑟𝑎𝑛𝑠𝑝𝑎𝑟𝑒𝑛𝑡 𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑐𝑒
Therefore:
Apply to generic derivation
sin 𝑖
𝑐1
=
sin 𝑟
𝑐2
Rearrange this formula to obtain properties of the first medium on the left hand side and properties of the second
medium on the right hand side.
Multiply both sides of the equation by c.
Substitute in our definition of the absolute refractive index for medium 1 and medium 2.
Snell’s Law of refraction
N.B. Approximations
The speed of light in air at atmospheric pressure is 99.97% speed of light in a vacuum. Calculate the refractive index
of air.
Thus what assumption can we make for the refractive index of air?