‹ Back to the paper
Derive the following.
For any prism, prove that: 'n' or μ = (sin((A + δ_m)/2))/(sin(A/2)) where the terms have their…
For any prism, prove that: $'n'\text{ or }\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ where the terms have their usual meaning.
To show: $\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
Answer
Answer
Refraction through a prism at minimum deviation.
- Let a ray PQ fall on face AB of a prism of angle $A$ at angle of incidence $i$, refract at angle $r_1$, travel along QR inside, and emerge from face AC at R at angle $e$ after refracting at angle $r_2$ inside; the deviation is $\delta$.
- Deviation at the two faces: $(i - r_1) + (e - r_2) = \delta$, so $\delta = i + e - (r_1 + r_2)$.
- In the quadrilateral AQNR (angles at Q and R are right angles), $A + \angle QNR = 180^\circ$, and in triangle QNR, $r_1 + r_2 + \angle QNR = 180^\circ$. Hence $r_1 + r_2 = A$.
- Substituting, $\delta = i + e - A$.
- At minimum deviation $\delta = \delta_m$ the ray passes symmetrically through the prism (QR parallel to the base), so $i = e$ and $r_1 = r_2 = r$.
- Then $2r = A$, i.e. $r = \frac{A}{2}$, and $\delta_m = 2i - A$, i.e. $i = \frac{A + \delta_m}{2}$.
- By Snell's law at the first face, $\mu = \frac{\sin i}{\sin r}$.
- Substituting the values of $i$ and $r$: $\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$.