Question

Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are μr, μy and μv respectively and those for the flint glass are μ'r, μ'y and μ'v respectively. Find the ratio A'/A for which (a) there is no net angular dispersion, and (b) there is no net deviation in the yellow ray.
Figure

Answer 1

The ratio A'/A for which (a) there is no net angular dispersion, and (b) there is no net deviation in the yellow ray has expressed.

Explanation:

In case of crown glass  :

Red rays Refractive index = μr

Yellow rays Refractive index  = μy

Violet rays Refractive index = μv

In case of flint glass :

Red rays Refractive index  = μ’r

Yellow rays Refractive index  = μ’y

Violet rays Refractive index  = μ’v

Let $\delta_{c y}$ and $\delta_f_y$ for yellow light, be the angles of deviation produced by the crown and flint prisms.

Maximum deviation provided for yellow rays by the combination of prism   :

$\delta y=\delta c y-\delta f y$

    $=2 \delta c y-\delta f y$

    $=2(\mu_c _y+1) A-\left(\mu_{f y}-1\right) A'$

The angular dispersion resulting from the combination is given by,$\delta_{v}-\delta_{r}=\left[\left(\mu_{v_{c}}-1\right) A-\left(\mu v_{f}-1\right) A^{\prime}+\left(\mu v_{c}-1\right) A ]-(\mu r c-1) A-(\mu r f-1) A^{\prime}+(\mu r c-1) A\right.$$\mu v_{c}=$ Refractive index for Crown glass violet colour

$\mu _v_{\mathrm{f}}=$ Violet colour of the flint glass Refractive index

$\mu_ r _c=$ Red colour of the crown glass Refractive index

$\mu _r _f=$ Red colour of the flint glass Refractive index  

$\delta_{\mathrm{v}}-\delta_{\mathrm{r}}=2\left(\mu_{\mathrm{vc}}-1\right) \mathrm{A}-\left(\mu_{\mathrm{vf}}-1\right) \mathrm{A}^{\prime}$

(a) We've got : No angular dispersion      

$\delta_{\mathrm{v}}-\delta_{\mathrm{r}}=0=2\left(\mu_{\mathrm{vc}}-1\right) \mathrm{A}-\left(\mu_{\mathrm{vf}}-1\right) \mathrm{A}^{\prime}$

$\frac{A^{\prime}}{A}=\frac{2\left(\mu_{v f}-1\right)}{\mu_{v c}-1}=\frac{2\left(\mu_{r}-1\right)}{\mu^{\prime} r-\mu}$

(b) For zero yellow-ray deviation, $\delta_{\mathrm{y}}$ = 0

$2\left(\mu_{\mathrm{cy}}-1\right) \mathrm{A}=(\mu_ \mathrm{fy}-1) \mathrm{A}$

$\frac{A^{\prime}}{A}=\frac{2\left(\mu_{c y}-1\right)}{\mu_{f y}-1}=\frac{2\left(\mu_{y}-1\right)}{\mu_{y}^{\prime}-1}$

Thus, the ratio A'/A for which (a) there is no net angular dispersion, and (b) there is no net deviation in the yellow ray has been derived.

Answer 2

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