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A ray of light passes through a prism of refractive index √2 as shown in the figure. Find:

(i) The angle of minimum deviation for this prism.

(ii) the angle of incidence (\(\angle r_2\)) at face AC.

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(i) The angle of minimum deviation

\(x = \frac{sin{\left(\frac{A+S_m}{2}\right)}}{sin\left(\frac A2\right)}\)

\(\sqrt2=\frac{sin\left(\frac{60 + S_m}{2}\right)}{sin\left(\frac{60}2\right)}\)

\(\sqrt2 \,sin\,30^\circ = sin\left(\frac{60^\circ+S_m}{2}\right)\)

\(\sqrt2 \times\frac12= sin\left(\frac{60^\circ+S_m}{2}\right)\)

\(\frac1{\sqrt2}= sin\left(\frac{60^\circ+S_m}{2}\right)\)

\(sin^{-1}\left(\frac1{\sqrt2}\right)= \frac{60^\circ+S_m}{2}\)

\(45^\circ= \frac{60^\circ+S_m}{2}\)

\(90^\circ= {60^\circ+S_m}\)

\(S_m = 30^\circ\)

(ii) Angle of incidence 

\(i = \frac{A+S_m}{2}\)

\(i = \frac{60^\circ+30^\circ}{2}\)

\(i = 45^\circ\)

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