Physics · Ch 6 — Optics

Refractive Index of the Material of the Prism

6.7.3

Refractive Index of the Material of the Prism

At minimum deviation, $i_1=i_2=i$ and $r_1=r_2=r$ ; substituting into $d=i_1+i_2-A$ gives $D=2i-A$ , i.e. $i=(A+D)/2$ , and substituting into $r_1+r_2=A$ gives $r=A/2$ . Substituting these into Snell's law $n=\sin i/\sin r$ gives the prism formula $\boxed{n=\dfrac{\sin[(A+D)/2]}{\sin(A/2)}}$ , letting a prism material's refractive index be determined purely from its apex angle $A$ and its measured minimum-deviati …

Misc Example 6.20Angle of deviation and refractive index from a grazing-emergence ray

Worked out. Reuses the same grazing-emergence scenario as the Critical Angle section's worked example: a ray at normal incidence (i1 = 0) on an equilateral prism (A = 60 degrees) emerges just grazing the second face (i2 = 90 degrees), giving deviation d = 0 + 90 - 60 = 30 degrees by the deviation formula. Because the internal ray strikes the second face at exactly the critical angle for this grazing condition, and applying sin(ic) = 1/n together with the prism's own geometry giving sin(ic) = sin(30 degrees) = 0.5, the refractive index of the prism material comes out t …

Misc Example 6.21Refractive index of a prism from its minimum deviation angle

Worked out. A prism of apex angle A = 60 degrees is measured to have a minimum deviation D = 37 degrees. Substituting into the prism formula n = sin[(A+D)/2]/sin(A/2) gives n = sin(97/2)/sin(30) = sin(48.5 degrees)/sin(30 degrees) = 0.75/0.5 = 1.5. This directly recovers the refractive index of the prism's material -- 1.5, a typical value for ordinary glass -- purely from two angles that can both be measured experimentally on a spectrometer, without needing to know anything else about the prism material …

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