question_answer

A glass prism of refracting angle \[60{}^\circ \] and refractive index 1.5, is completely immersed in water of refractive index 1.33. Calculate the angle of minimum deviation of the prism in this situation.

\[(given,si{{n}^{-1}}0.56=34.3{}^\circ )\]

Or

(i) The refractive indices of crown glass for red and violet colours are 1.515 and 1.523 respectively. Find the dispersive power of crown glass between these colours.

(ii) If a crown glass prism produces a mean deviation of \[40{}^\circ ,\] what will be the angular dispersion?

Answer:

We know that,    \[{}^{w}{{\mu }_{g}}=\frac{\sin \left( \frac{A+{{\delta }_{m}}}{2} \right)}{\sin \left( \frac{A}{2} \right)}\] \[\frac{{}^{a}{{\mu }_{g}}}{{}^{a}{{\mu }_{w}}}=\frac{\sin \left( \frac{A+{{\delta }_{m}}}{2} \right)}{\sin (A/2)}\]             \[\Rightarrow \]               \[\frac{1.5}{1.33}=\frac{\sin \left( \frac{60{}^\circ +{{\delta }_{m}}}{2} \right)}{\sin (60{}^\circ /2)}\]             or         \[\sin \left( \frac{60{}^\circ +{{\delta }_{m}}}{2} \right)=\frac{1.5}{1.33}\times \frac{1}{2}=0.56\]             or         \[\left( \frac{A+{{\delta }_{m}}}{2} \right)={{\sin }^{-1}}(0.56)=34.3{}^\circ \]             \[\Rightarrow \] \[A+{{\delta }_{m}}=68.6{}^\circ \] or \[{{\delta }_{m}}=68.6{}^\circ -60{}^\circ =8.6{}^\circ \] (i) The dispersive power of crown glass is given by                         \[\omega =\frac{{{\mu }_{V}}-{{\mu }_{R}}}{{{\mu }_{Y}}-1}\] Given,   \[{{\mu }_{V}}=1.523;{{\mu }_{R}}=1.515\] So,       \[{{\mu }_{Y}}=\frac{{{\mu }_{V}}+{{\mu }_{R}}}{2}=\frac{1.523+1.515}{2}\]             = 1.519 \[\therefore \]      \[\omega =\frac{1.523-1.515}{1.519-1}=\frac{0.008}{0.519}=0.0154\] (ii) If \[{{\delta }_{V}},{{\delta }_{R}}\] and \[{{\delta }_{Y}}\] be the deviations produced by the crown-glass prism in violet, red and yellow, respectively rays of light, then the dispersive power of crown glass between red and violet rays is             \[\omega =\frac{{{\delta }_{V}}-{{\delta }_{R}}}{{{\delta }_{Y}}}\] \[\therefore \] Angular dispersion between red and violet rays is             \[\theta ={{\delta }_{V}}-{{\delta }_{R}}=\omega \times {{\delta }_{Y}}\]             \[=0.0154\times 40{}^\circ =0.616{}^\circ \]