Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are $\mu_\text{r},\mu_\text{y}$ and $\mu_\text{v}$ respectively and those for the flint glass are $\mu'_\text{r},\mu'_\text{y}$ and $\mu'_\text{v}$ respectively. Find the ratio $\frac{\text{A}'}{\text{A}}$ for which.
- There is no net angular dispersion.
- There is no net deviation in the yellow ray.
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Total deviation for yellow ray produced by the prism combination is,
$\delta_\text{y}=\delta_\text{cy}-\delta_\text{fy}+\delta_\text{cy}=2\delta_\text{cy}-\delta_\text{fy}$ $=2(\mu_\text{cy}-1)\text{A}-(\mu_\text{cy}-1)\text{A}'$
Similarly the angular dispersion produced by the combination is
$\delta_\text{v}-\delta_\text{r}=\big[(\mu_\text{vc}-1)\text{A}-(\mu_\text{vf}-1)\text{A}'+(\mu_\text{vc}-1)\text{A}\big]\\\big[(\mu_\text{rc}-1)\text{A}-(\mu_\text{rf}-1)\text{A}'+(\mu_\text{r}-1)\text{A})\big]$
$=2(\mu_\text{vc}-1)\text{A}-(\mu_\text{vf}-1)\text{A}'$
$\Rightarrow2(\mu_\text{vc}-1)\text{A}=(\mu_\text{vf}-1)\text{A}'$
$\Rightarrow\frac{\text{A}'}{\text{A}}=\frac{2(\mu_\text{cv}-\mu_\text{rc})}{(\mu_\text{vf}-\mu_\text{rf}}$
$=\frac{2(\mu_\text{v}-\mu_\text{r})}{(\mu'_\text{v}-\mu'_\text{r})}$
$\Rightarrow2(\mu_\text{cy}-1)\text{A}=(\mu_\text{fy}-1)\text{A}'$
$\Rightarrow\frac{\text{A}'}{\text{A}}=\frac{2(\mu_\text{cy}-1)}{(\mu_\text{fy}-1)}$
$=\frac{2(\mu_\text{y}-1)}{(\mu'_\text{y}-1)}$
$\delta_\text{y}=\delta_\text{cy}-\delta_\text{fy}+\delta_\text{cy}=2\delta_\text{cy}-\delta_\text{fy}$ $=2(\mu_\text{cy}-1)\text{A}-(\mu_\text{cy}-1)\text{A}'$
Similarly the angular dispersion produced by the combination is
$\delta_\text{v}-\delta_\text{r}=\big[(\mu_\text{vc}-1)\text{A}-(\mu_\text{vf}-1)\text{A}'+(\mu_\text{vc}-1)\text{A}\big]\\\big[(\mu_\text{rc}-1)\text{A}-(\mu_\text{rf}-1)\text{A}'+(\mu_\text{r}-1)\text{A})\big]$
$=2(\mu_\text{vc}-1)\text{A}-(\mu_\text{vf}-1)\text{A}'$
- For net angular dispersion to be zero,
$\Rightarrow2(\mu_\text{vc}-1)\text{A}=(\mu_\text{vf}-1)\text{A}'$
$\Rightarrow\frac{\text{A}'}{\text{A}}=\frac{2(\mu_\text{cv}-\mu_\text{rc})}{(\mu_\text{vf}-\mu_\text{rf}}$
$=\frac{2(\mu_\text{v}-\mu_\text{r})}{(\mu'_\text{v}-\mu'_\text{r})}$
- For net deviation in the yellow ray to be zero,
$\Rightarrow2(\mu_\text{cy}-1)\text{A}=(\mu_\text{fy}-1)\text{A}'$
$\Rightarrow\frac{\text{A}'}{\text{A}}=\frac{2(\mu_\text{cy}-1)}{(\mu_\text{fy}-1)}$
$=\frac{2(\mu_\text{y}-1)}{(\mu'_\text{y}-1)}$
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