Question
Derive $D_m=A\left(n_{21}-1\right)$ for thin lens.
✓
Answer
$\rightarrow$ Refractive index of prism $n_{21}=\frac{\sin \left(\frac{A+D}{2}\right)}{\sin \frac{A}{2}} \ldots(1)$
$\rightarrow$ But for thin prism, prism angle $A$ is small and,
$\therefore \sin \left(\frac{ A + D _m}{2}\right) \approx \frac{ A + D _m}{2}$
$ \sin \left(\frac{ A }{2}\right) \approx \frac{ A }{2}$
$\rightarrow$ Substituting in equation $(1),$
$\therefore n_{21}=\frac{\frac{A+D_m}{2}}{\frac{A}{2}}$
$\therefore n_{21}=\frac{ A + D _m}{A}$
$\therefore n_{21} \cdot A = A + D _m$
$\therefore D _m=n_{21} \cdot A - A$
$\therefore D _m= A \left(n_{21}-1\right)$
$\rightarrow$ From this formula it can be said that thin prism $($Thin prism means $' A \ '$ will be smaller and hence $D _{ m }$ will be smaller, too.$)$ does not deflect light much.
$\rightarrow$ But for thin prism, prism angle $A$ is small and,
$\therefore \sin \left(\frac{ A + D _m}{2}\right) \approx \frac{ A + D _m}{2}$
$ \sin \left(\frac{ A }{2}\right) \approx \frac{ A }{2}$
$\rightarrow$ Substituting in equation $(1),$
$\therefore n_{21}=\frac{\frac{A+D_m}{2}}{\frac{A}{2}}$
$\therefore n_{21}=\frac{ A + D _m}{A}$
$\therefore n_{21} \cdot A = A + D _m$
$\therefore D _m=n_{21} \cdot A - A$
$\therefore D _m= A \left(n_{21}-1\right)$
$\rightarrow$ From this formula it can be said that thin prism $($Thin prism means $' A \ '$ will be smaller and hence $D _{ m }$ will be smaller, too.$)$ does not deflect light much.
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