**1.** 1. A flint glass prism and a crown glass prism are to be
combined in such a way that the deviation of the mean
ray is zero. The refractive index of flint and crown
glasses for the mean ray are 1'620 and 1'518
respectively. If the refracting angle of the flint prism is
6.0°, what would be the refracting angle of the crown
prism ?

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SolutionsGiven that $\\$ Refractive index of flint glass = $\mu_f=1.620 $ $\\$ Refractive index of crown glass= $\mu_c=1.528 $ $\\$ Refracting angle of flint prism = $A_f$=$6.0^{\circ} $$\\$ For zero net deviation of mean ray $\\$ ($\mu_1-1)A_f=(\mu_c-1)A_c $ $\\$ $\Rightarrow A_c=\frac{\mu_c -1}{\mu_c-1}A_f=\frac{1.620-1}{1.518-1}(6.0)^{\circ}=7.2^{\circ} $ $\\$

**2.** 2. A certain material has refractive indices 1'56, 1'60 and
1'68 for red, yellow and violet light respectively.
(a) Calculate the dispersive power. (b) Find the angular
dispersion produced by a thin prism of angle 6° made of
this material.

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SolutionsGiven that $\mu_r=1.56, \mu_y = 1.60 and \mu_v=1.68 $ $\\$ a)Dispersive power = $\omega=\frac{\mu_v-\mu_r}{\mu_y-1}=\frac{1.68-1.56}{1.60-1}=0.2 $ $\\$ b)Angular dispersion =($\mu_v-\mu_r)A=0.12\times6^{\circ}=7.2^{\circ} $ $\\$

**3.** 3. The focal lengths of a convex lens for red, yellow and
violet rays are 100 cm, 98 cm and 96 cm respectively.
Find the dispersive power of the material of the lens.

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SolutionsThe focal length of a lens is given by $\\$ $\frac{1}{f}=(\mu-1)(\frac{1}{R_1}-\frac{1}{R_2}) $ $\\$ $\Rightarrow (\mu-1)=\frac{1}{f}\times\frac{1}{(\frac{1}{R_1}-\frac{1}{R_2})} =\frac{K}{f} ..(1) ..(1) $ $\\$ So $\mu_r-1=\frac{K}{100}$ ..(2) $\\$ $\mu_y-1=\frac{k}{98} ..$(3) $\\$ And $ \mu_v-1=\frac{K}{96} $ (4)$\\$ So dispersive power = $\omega=\frac{\mu_v-\mu_r}{\mu_y-1}=\frac{(\mu_v-1)-(\mu_r-1)}{\mu_y-1}=\frac{\frac{k}{96}-\frac{k}{100}}{\frac{k}{98}}=\frac{\frac{k}{96}-\frac{k}{100}}{\frac{k}{98}}=\frac{98\times4}{9600}=0.0408$ $\\$

**4.** 5. A thin prism is made of a material having refractive
indices 1'61 and 1'65 for red and violet light. The
dispersive power of the material is 0'07. It is found that
a beam of yellow light passing through the prism suffers
a minimum deviation of 4'0° in favourable conditions.
Calculate the angle of the prism.

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SolutionsGiven that $\mu_r=1.61,\hspace{0.3cm}\mu_v=1.65, \omega=0.07\hspace{0.2cm} and\hspace{0.2cm} \delta_y=4^{\circ} $ $\\$ Now, $\omega =\frac{\mu_v-\mu_r}{\mu_y-1} $ $\\$ $\Rightarrow 0.07=\frac{1.65-1.61}{\mu_y-1} $ $\\$ $\Rightarrow \mu_y-1=\frac{0.04}{0.07} =\frac{4}{7} $ $\\$ Again, $\delta=(\mu-1)A $ $\\$ $\Rightarrow A= \frac{\delta_y}{\mu_y-1}=\frac{4}{(4/7)}=7^{\circ} $ $\\$

**5.** 4. The refractive index of a material changes by 0'014 as
the colour of the light changes from red to violet. A
rectangular slab of height 2'00 cm made of this material
is placed on a newspaper. When viewed normally in
yellow light, the letters appear 1'32 cm below the top
surface of the slab. Calculate the dispersive power of the
material.

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SolutionsGiven that $\mu_v-\mu_r=0.014$ $\\$ Again, $\mu_y=\frac{Real Depth}{Apparent depth}=\frac{2.00}{1.300}=1.515 $ $\\$ So, dispersive power =$\frac{\mu_v-\mu_r}{\mu_y-1}=\frac{0.014}{1.515-1}=0.027 $ $\\$

**6.** 6. The minimum deviations suffered by red, yellow and
violet beams passing through an equilateral transparent
prism are 38'4°, 38'7° and 39'2° respectively. Calculate
the dispersive power of the medium.

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SolutionsGiven that $\delta_r=38.4^{\circ},\delta_y=38.7^{\circ} and \hspace{0.2cm}\delta_v=39.2^{\circ} $ $\\$ Dispersive power =$\frac{\mu_v-\mu_r}{\mu_y-1}=\frac{(\mu_v-1)-(\mu_r-1)}{(\mu_y-1)}=\frac{(\frac{\delta_v}{A})-(\frac{\delta_r}{A})}{(\frac{\delta_v}{A})} $ $\\$

**7.** 7. Two prisms of identical geometrical shape are combined
with their refracting angles oppositely directed. The
materials of the prisms have refractive indices 1'52 and
1'62 for violet light. A violet ray is deviated by 1'0° when
passes symmetrically through this combination. What is
the angle of the prisms ?

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SolutionsTwo prisms of identical geometrical shape are combined. $\\$ Let A = Angle of the prisms $\\$ $\mu'_v=1.52 and\mu_v=1.62,\delta_v=1^{\circ} $ $\\$ $\delta_v=(\mu_v-1)A-(\mu'_v-1)A [Since A=A'] $ $\\$ $\Rightarrow A=\frac{\delta_v}{\mu_v-\mu'_v}=\frac{1}{1.62-1.52}=10^{\circ} $ $\\$

**8.** 7. Two prisms of identical geometrical shape are combined
with their refracting angles oppositely directed. The
materials of the prisms have refractive indices 1'52 and
1'62 for violet light. A violet ray is deviated by 1'0° when
passes symmetrically through this combination. What is
the angle of the prisms ?

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Solutions**9.** 8. Three thin prisms are combined as shown in figure
. The refractive indices of the crown glass for red,
yellow and violet rays are $\mu_r$ , $\mu_s$ and $\mu_y$, respectively and those for the flint glass are $\mu'_r$,$ \mu'_y $and $\mu's$, 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.

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SolutionsTotal deviation for yellow ray produced by the prism combination is $\\$ $ \delta_y=\delta_{cy}-\delta_{fy}+\delta_{cy}=2\delta_{cy}-\delta_{fy}=2(\mu_cy-1)A-(\mu_cy-1)A' $ $\\$ Similarly, the angular dispersion produced by the combination is $\\$ $\delta_v-\delta_r=[(\mu_{vc}-1)A-(\mu_{vf}-1)A'+(\mu_{vc}-1)A]-[(\mu_{rc}-1)A-(\mu_d-1)A'+(\mu_r-1)A)] $ $\\$ $=2(\mu{vc}-1)A-(\mu_{Vf}-1)A' $ $\\$ $\Rightarrow \frac{A'}{A}=\frac{2(\mu_{cv}-\mu_{rc})}{(\mu_{vf}-\mu_{rf})}=\frac{2(\mu_V-\mu_r)}{(\mu'_v-\mu'_r)}$

**10.** 9. A thin prism of crown glass ($\mu_r$, = 1.515, $\mu_s$, = 1'525) and
a thin prism of flint glass ($\mu_r$, = 1'612,. $\mu_s$ = 1'632) are
placed in contact with each other. Their refracting angles
are 5'0° each and are similarly directed. Calculate the
angular dispersion produced by the combination.

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SolutionsGiven that $\mu_{cr}$=1.515,$\mu_{cv}$=1.525 and $\mu_{fr}$=1.612,$\mu_{fv}$=1.632 and A=$5^{\circ}. $ $\\$ Since, they are similarly detected, the total deviation produced is given by, $\\$ $\delta=\delta_c+\delta_r=(\mu_c-1)A+(\mu_r-1)A=(\mu_c+\mu_r-2)A $ $\\$ So, the angular dispersion of the combination is given by, $\\$ $\delta_v-\delta_y=(\mu_{cv}+\mu_fv-2)A-(\mu_{cr}+\mu_{fr}-2)A $ $\\$ =$(\mu_{cv} +\mu_{fv}-\mu{cr}-\mu{fr})A=(1.525 + 1.632-1.515-1.612)5=0.15^{\circ} $ $\\$

**11.** 10. A thin prism of angle 6.0°,$\omega$ = 0.07 and $\mu_y$ = 1.50 is
combined with another thin prism having $\omega$= 0.08 and
$\mu_y$ = 1.60. The combination produces no deviation in the
mean ray. (a) Find the angle of the second prism. (b)
Find the net angular dispersion produced by the
combination when a beam of white light passes through
it. (c) If the prisms are similarly directed, what will be
the deviation in the mean ray ? (d) Find the angular
dispersion in the situation described in (c).

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SolutionsGiven that$ A^{\circ}=6,\hspace{0.3cm}\omega'=0.07 \hspace{0.25cm} \mu'y=1.50$ $\\$ A=? $\hspace{0.5cm} \omega=0.08 \hspace{0.3cm} \mu y=1.60 $ $\\$ The combination produces no deviation in the mean ray $\\$ (a) $\delta_y=(\mu_y-1)A-(\mu'_y-1)A'=0$ [prism must be oppositely directed] $\Rightarrow (1.60-1)A=((1.50-1)A' $ $\\$ $\Rightarrow A=\frac{0.50\times6^{\circ}}{0.60}=5^{\circ} $ $\\$ (b) When a beam of white light passes throught it, $\\$ Net angular dispersion =$ (\mu_y-1)\omega A-(\mu'_y-1)\omega' A' $ $\\$ $\Rightarrow(1.60-1)(0.08)(5^{\circ})-(1.50-1)(0.07)(6^{\circ})$ $\\$ $\Rightarrow 0.24^{\circ}-0.21^{\circ}=0.03^{\circ} $ $\\$ (c) If the prisms are similarly directed, $\delta_y=(\mu_y-1)A+(\mu'_y-1)A $ $\\$ =(1.60-1)$5^{\circ}$+(1.50-1)$6^{\circ}$=$3^{\circ}+3^{\circ}=6^{\circ} $ $\\$ (d) Similarly, if the prisms are similarly directed , the net angular dispersion is given by, $\\$ $ \delta_v-\delta_r=(\mu_y-1)\omega A -(\mu'_y-1)\omega'A'=0.24^{\circ}+0.21^{\circ}=0.45^{\circ} $ $\\$

**12.** 11. The refractive index of a material M1 changes by 0'014
and that of another material M2 changes by 0.024 as
the colour of the light is changed from red to violet. Two
thin prisms one made of M1(A = 5.3°) and other made
of M2(A = 3.7°) are combined with their refracting angles
oppositely directed. (a) Find the angular dispersion
produced by the combination. (b) The prisms are now
combined with their refracting angles similarly directed.
Find the angular dispersion produced by the
combination.

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SolutionsGiven that $\mu'_v-\mu'_r=0..014 and \mu_v-\mu_r=0.024 $ $\\$ $A'=5.3^{\circ} and A=3.7^{\circ} $ $\\$ (a) When the prisms are oppositely directed $\\$ angular dispersion = $(\mu_v-\mu_r)A-(\mu'_v-\mu'_r)A' $ $\\$ =0.024$\times$3.7$^{\circ}+0.014\times5.3^{\circ}=0.163^{\circ} $ $\\$