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 ratio $\frac{A^{\prime }}{A}$ for which (a) there is no net angular dispersion, and (b) there is no net deviation in the yellow ray.

Answer 1

Total deviation for yellow ray produced by the prismcombination is,

${\delta }_y={\delta }_{cy}-{\delta }_{ty}$

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

$=2({\mu }_y+1)A-({\mu }_{rc}-1)A$

Similarly the angular dispersion produced by the combination is

$\delta {\mu }_r-{\delta }_r=[{\mu }_{vc}-1)A-({\mu }_{vt}-1)A+({\mu }_{vc}-1)A-({\mu }_{rt}-1)A(\mu -1)A]$

$=2({\mu }_{vc}-{\mu }_{rc})A-({\mu }_{vt}-{\mu }_{rt})A$

(a) For net angular dispersion to be zero,

${\delta }_t-{\delta }_t=0=2({\mu }_{vc}-{\mu }_{tc})A$

$=({\mu }_{vt}-{\mu }_{rt})A$

$\Rightarrow \frac{A}{A}=\frac{2({\mu }_{vc}-{\mu }_{rc})}{({\mu }_{vc}-{\mu }_{rc})}$

$=\frac{2({\mu }_r-{\mu }_r)}{({\mu }_r-\mu )}$

(b) For net deviation in the yellow ray to be zero,

$\delta =0$

$\Rightarrow 2({\mu }_{cy}-1)A=({\mu }_{ty}-1)A$

$\Rightarrow \frac{A}{A}=\frac{2({\mu }_{cy}-1)}{({\mu }_{ty}-1)}$

$=\frac{2({\mu }_y-1)}{({\mu }_y-1)}$