Derive n=sin(A+D/2)/sinA/2 for refraction through a prism.(With usual notations) ​

Question

Derive n= \(\frac{sin(\frac{A+D}{2})}{sin A/2}\) for refraction through a prism.(With usual notations)

Answer 1

A ray of light PO incident on the face AB of a glass prism ABC of angle A and refractive index 'n'.

In quadrilateral AOMO¹

\(\angle A + \angle M = 180^0\) ...............(1)

In \(\triangle^{le} OMO^1\)

\(\angle r_1 + \angle r_2 + \angle M = 180^0\) ................(2)

From (1) and (2)

\(\angle A + \angle M = \angle r_1 + \angle r_2 + \angle M\)

\(\angle A = \angle r_1 + \angle r_2\) ............................(3)

In \(\triangle SOO^1\),

\(\angle d = (i_1-r_1)+(i_2-r_2)\)

= \((i_1+i_2)-(r_1+r_2)\)

= \((i_1+i_2)-(A)\)

A+d= \(i_1+i_2\) .............................(4)

But,. \(i_1=i_2\), \(r_1=r_2\)

At minimum, deviation position, d=D and \(i_1=i_2=i\) and \(r_1=r_2=r\)

Equations (3) and (4) becomes

A = \(\angle r_1 + \angle r_2\) = r + r = 2r

r = A/2

A + D = \(i_1+i_2\) = i + i = 2i

i = \(\frac{A+D}{2}\)

Substituting the values of i and r in the Snell's law equation. We get

n = \(\frac{sin\ i}{sin\ r}\)

n= \(\frac{sin(A+D)/2}{sinA/2}\)