Show that the refractive index, n, of a medium is given by n = (real depth)/(apparent depth)

Show that the refractive index, n, of a medium is given by n  = (real depth/apparent depth)

Solution

Consider an object O viewed normally from above through a parallel-sided glass block of refractive index ng and thickness t as shown.

A ray from an object O normal to the glass surface at M passes undeviated. While the ray ON inclined at a small angle i to the normal is refracted at N toward P. The observer above the glass block sees the image of the object O at I.

Applying Snell’s law at N

n_gsin i = n_a sin r …………………………….. (i)

From the diagram; sin i = MN/ON  and sin r =MN/IN

Substituting in equation (i);  n_g(MN/ON) = n_a(MN/IN)

n_a= 1 then    n_g = ON/IN

Since angle i is very small, then ON ≈  OM and IN ≈  IM

n_g = OM/IM

From the diagram, OM = real depth and IM = apparent depth.

Hence, refractive index, n  = (real depth/apparent depth)

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