Qstnid -Iv-8-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 (A) There Is No Net Angular Dispersion, And (B) There Is No Net Deviation In The Yellow Ray. figure - 20: Dispersion And Spectra

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NEET-XII-Physics

20: dispersion and Spectra

with Solutions - page 4

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#8
Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are μ_r, μ_y and μ_v 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 (a) there is no net angular dispersion, and (b) there is no net deviation in the yellow ray.
Figure
Ans : For the crown glass, we have:
Refractive index for red rays = μ_r
Refractive index for yellow rays = μ_y
Refractive index for violet rays =μ_v
For the flint glass, we have:
Refractive index for red rays = μ'_r
Refractive index for yellow rays = μ'_y
Refractive index for violet rays =μ'_v
Let δ_cyand δ_fybe the angles of deviation produced by the crown and flint prisms for the yellow light.
Total deviation produced by the prism combination for yellow rays:
δ_y = δ_cy - δ_fy
= 2δ_cy - δ_fy
=2(μ_cy + 1)A - (μ_fy - 1)A'
Angular dispersion produced by the combination is given by
δ_v - δ_r = [(μ_vc - 1)A - (μ_vf - 1)A' + (μ_v_c - 1)A - `` \left[\left({\mu }_{rc}-1\right)A-\left({\mu }_{rf}-1\right)A\text{'}+\left({\mu }_{rc}-1\right)A\right]``
Here,
μ_vc₌Refractive index for the violet colour of the crown glass
μ_vf = Refractive index for the violet colour of the flint glass
`` {\mu }_{rc}`` = Refractive index for the red colour of the crown glass
`` {\mu }_{rf}`` = Refractive index for the red colour of the flint glass
On solving, we get:
δ_v - δ_r = 2(μ_vc -1)A - (μ_vf - 1)A' (a) For zero angular dispersion, we have:
δ_t - δ_t = 0 = 2(μ_vc -1)A - (μ_vf - 1)A'
`` \Rightarrow \frac{A\text{'}}{A}=\frac{2({\mu }_{\,\mathrm{\,vf\,}}-1)}{({\mu }_{vc}-1)}``
`` =\frac{2({\mu }_{r}-{\mu }_{r})}{({\mu }_{r}-\mu )}`` (b) For zero deviation in the yellow ray, δ_y = 0.
⇒ 2(μ_cy - 1)A = (μ_fy - 1)A
`` \Rightarrow \frac{A\text{'}}{A}=\frac{2({\mu }_{\,\mathrm{\,cy\,}}-1)}{({\mu }_{\,\mathrm{\,fy\,}}-1)}``
`` =\frac{2({\mu }_{y}-1)}{(\mu {\text{'}}_{y}-1)}``
Page No 442:
Page No 443: (a) For zero angular dispersion, we have:
δ_t - δ_t = 0 = 2(μ_vc -1)A - (μ_vf - 1)A'
`` \Rightarrow \frac{A\text{'}}{A}=\frac{2({\mu }_{\,\mathrm{\,vf\,}}-1)}{({\mu }_{vc}-1)}``
`` =\frac{2({\mu }_{r}-{\mu }_{r})}{({\mu }_{r}-\mu )}`` (b) For zero deviation in the yellow ray, δ_y = 0.
⇒ 2(μ_cy - 1)A = (μ_fy - 1)A
`` \Rightarrow \frac{A\text{'}}{A}=\frac{2({\mu }_{\,\mathrm{\,cy\,}}-1)}{({\mu }_{\,\mathrm{\,fy\,}}-1)}``
`` =\frac{2({\mu }_{y}-1)}{(\mu {\text{'}}_{y}-1)}``
Page No 442:
Page No 443:

#8-a
there is no net angular dispersion, and
Ans : For zero angular dispersion, we have:
δ_t - δ_t = 0 = 2(μ_vc -1)A - (μ_vf - 1)A'
`` \Rightarrow \frac{A\text{'}}{A}=\frac{2({\mu }_{\,\mathrm{\,vf\,}}-1)}{({\mu }_{vc}-1)}``
`` =\frac{2({\mu }_{r}-{\mu }_{r})}{({\mu }_{r}-\mu )}``

#8-b
there is no net deviation in the yellow ray.
Figure
Ans : For zero deviation in the yellow ray, δ_y = 0.
⇒ 2(μ_cy - 1)A = (μ_fy - 1)A
`` \Rightarrow \frac{A\text{'}}{A}=\frac{2({\mu }_{\,\mathrm{\,cy\,}}-1)}{({\mu }_{\,\mathrm{\,fy\,}}-1)}``
`` =\frac{2({\mu }_{y}-1)}{(\mu {\text{'}}_{y}-1)}``
Page No 442:
Page No 443: