A ray of light is incident on a surface of glass slab at an angle \(45^{\circ}\). If the lateral shift produced per unit thickness is \(\frac{1}{\sqrt{3}} \mathrm{~m}\), the angle of refraction produced is

Here, angle of incidence \(i=45^{\circ}\)



\(\frac{\text { Lateral shift }(\mathrm{d})}{\text { Thickness of glass slab (t) }}=\frac{1}{\sqrt{3}}\)
Lateral shift \(\mathrm{d}=\frac{\mathrm{t} \sin \delta}{\cos \mathrm{r}}=\frac{\mathrm{t} \sin (\mathrm{i}-\mathrm{r})}{\cos \mathrm{r}}\)
\(\Rightarrow \quad \frac{\mathrm{d}}{\mathrm{t}}=\frac{\sin (\mathrm{i}-\mathrm{r})}{\cos \mathrm{r}}\)
or \(\quad \frac{\mathrm{d}}{\mathrm{t}}=\frac{\sin \mathrm{i} \cos \mathrm{r}-\cos \mathrm{i} \sin \mathrm{r}}{\cos \mathrm{r}}\)
or \(\quad \frac{\mathrm{d}}{\mathrm{t}}=\frac{\sin 45^{\circ} \cos \mathrm{r}-\cos 45^{\circ} \sin \mathrm{r}}{\cos \mathrm{r}}\)
\(=\frac{\cos \mathrm{r}-\sin \mathrm{r}}{\sqrt{2} \cos \mathrm{r}}\)
or \(\quad \frac{\mathrm{d}}{\mathrm{t}}=\frac{1}{\sqrt{2}}(1-\tan \mathrm{r})\)
or \(\quad \frac{1}{\sqrt{3}}=\frac{1}{\sqrt{2}}(1-\tan \mathrm{r})\)
or \(\quad \tan \mathrm{r}=1-\frac{\sqrt{2}}{\sqrt{3}}\)
or \(\quad \tan { }^{-1}\left(1-\frac{\sqrt{2}}{\sqrt{3}}\right)\)