Question

Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are ${\mu }_r,$ ${\mu }_y$ and ${\mu }_v$ respectively and those for the flint glass are ${\mu }_r^{{}^{\prime }}$, ${\mu }_y^{{}^{\prime }}$ and ${\mu }_v^{{}^{\prime }}$ respectively. Find the $\frac{A^{\prime }}{A}$ for which, there is no net angular dispersion.

• A. $\frac{{\mu }_y-1}{{\mu }_y^{{}^{\prime }}-1}$
• B. $\frac{2{\mu }_y-1}{{\mu }_y^{{}^{\prime }}-1}$
• C. $\frac{3{\mu }_y-1}{{\mu }_y^{{}^{\prime }}-1}$
• D. $\frac{4{\mu }_y-1}{{\mu }_y^{{}^{\prime }}-1}$

Answer 1

The correct option is B $\frac{2{\mu }_y-1}{{\mu }_y^{{}^{\prime }}-1}$

Total deviation for yellow ray produced

$\delta y={\delta }_{cy}-{\delta }_{fy}+{\delta }_{cy}$

$=2{\delta }_{cy}-{\delta }_{fy}$

$=2\left({\mu }_{cy-1}\right)A-({\mu }_{-cy}-1)A^1$

Angular dispersion,

${\delta }_v-{\delta }_r=\left[\right({\mu }_{vc}-1)A-({\mu }_{vf}-1)A^1+({\mu }_{vc}-1\left)A\right]$

$-\left[\right({\mu }_{rc}-1)A-({\mu }_{rf}-1)A^1+({\mu }_r-1\left)A\right]$

$2({\mu }_{vc}-1)A-({\mu }_{vf}-1)A^1$

For net angular dispersion to be zero,

${\delta }_r-{\delta }_r=0$

$2({\mu }_{vc}-1)A=({\mu }_{vf}-1)A^1$