Q.Derive the equation for angle of deviation produced by a prism and thus obtain the equation for the refractive index of the material of the prism.

Step 1. A ray refracts at the first face of a prism (incidence $i_1$, refraction $r_1$, deviation $d_1=i_1-r_1$) and at the second face (internal incidence $r_2$, emergence $i_2$, deviation $d_2=i_2-r_2$); the total deviation is $d=d_1+d_2=(i_1+i_2)-(r_1+r_2)$.

Step 2. In quadrilateral $AQNR$ (formed by the apex $A$ and the two normals at the points of refraction $Q,R$, meeting at $N$), two angles (at $Q$ and $R$) are right angles, so the other two angles sum to $180°$: $\angle A+\angle QNR=180°$.

Step 3. In triangle $QNR$, $r_1+r_2+\angle QNR=180°$. Comparing with Step 2, $\angle QNR=180°-A$, so $r_1+r_2+180°-A=180°$, giving $r_1+r_2=A$.

Step 4. Substituting into Step 1: $\boxed{d=i_1+i_2-A}$. …

Derive d = i1+i2-A from the quadrilateral/triangle geometry at the two refracting faces, then special …

• Forgetting to prove r1+r2=A from the quadrilateral geometry before substituting into the deviation formula. …