For a small-angled prism,derive the expression for the angle of minimum deviation,${D_m} = A({n_{21}} - 1)$.

(N/A) For a prism,the angle of deviation $D$ is given by $D = i + e - A$,where $A$ is the prism angle,$i$ is the angle of incidence,and $e$ is the angle of emergence.
At the position of minimum deviation,$i = e$ and $r_1 = r_2 = r = A/2$.
Using Snell's law at the first surface,$n_1 \sin i = n_2 \sin r_1$.
For a small-angled prism,$i$ and $r_1$ are small,so $\sin i \approx i$ and $\sin r_1 \approx r_1$.
Thus,$n_1 i = n_2 r_1$,which implies $i = (n_2/n_1) r_1 = n_{21} (A/2)$.
Substituting $i = e$ into the deviation formula: $D_m = i + i - A = 2i - A$.
Substituting $i = n_{21} (A/2)$ into the equation: $D_m = 2(n_{21} \cdot A/2) - A$.
Simplifying,we get $D_m = n_{21} A - A = A(n_{21} - 1)$.