Tuesday, July 30, 2019

H C Verma solutions, Dispersion and Spectra, Exercises, Chapter-20, Concepts of Physics, Part-I

Dispersion and Spectra

EXERCISES


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?                   


ANSWER: The net deviation of the mean ray in this combination is given by,

  ẟ = (µᵧ - 1)A - (µ'ᵧ - 1)A'.

Here given that, ẟ = 0, µᵧ = 1.620, µ'ᵧ = 1.518, A = 6.0°, A' = ? Putting the values

0 = (1.620 - 1)A -(1.518 - 1)A'

→0.518A' = 0.620*A

→A' = 1.2*A = 1.2*6° = 7.2°         

 


  

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.                   


ANSWER:  Given, µᵣ = 1.56, µᵧ = 1.60 and µᵥ = 1.68.

(a) The dispersive power ⍵ = (µᵥ - µᵣ)/(µᵧ - 1)

→⍵ = (1.68 - 1.56)/(1.60 - 1)

→⍵ = 0.12/0.60 = 0.2 


(b) The angular dispersion is

ẟᵥ - ẟᵣ = (µᵥ - µᵣ)A = (1.68 - 1.56)*6° =0.72°      

   


 

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.                    


ANSWER:  From lens makers formula, the focal length of a lens is given as

1/f = (µ-1){1/R₁-1/R₂} = K(µ-1)

Where K = {R₁-1/R₂} =constant

→µ-1 = 1/Kf

So, µᵥ - 1 = 1/Kfᵥ

      µᵣ - 1 = 1/Kfᵣ 

      µᵧ - 1 = 1/Kfᵧ 

Now, (µᵥ-µᵣ) =(µᵥ - 1) - (µᵣ - 1) = (1/K){1/fᵥ - 1/fᵣ}     

Hence the dispersive power of the material of the lens, ⍵ =(µᵥ-µᵣ)/(µᵧ-1)

→⍵ = {1/fᵥ - 1/fᵣ}/(1/fᵧ)

Given fᵥ = 96 cm, fᵣ = 100 cm, fᵧ = 98 cm

Hence ⍵ = {1/96 - 1/100}/(1/98)
             = {(100-96)/9600}/(1/98)
             = 4*98/9600
             = 0.041

   


 

4. The refractive index of a material changes by 0.014 as the color 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.                    


ANSWER:  Given, µᵥ - µᵣ = 0.014

The thickness of the rectangular slab = The real depth of the newspaper = 2.00 cm

Apparent depth of the newspaper in yellow light = 1.32 cm

Hence µᵧ = Real depth/Apparent depth = 2.00/1.32 = 1.52

µᵧ - 1 = 1.52 - 1 = 0.52

Hence the dispersive power of the material

⍵ = (µᵥ - µᵣ)/(µᵧ - 1) = 0.014/0.52 = 0.026          

   

 

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 favorable conditions. Calculate the angle of the prism.                   


ANSWER:  Given that, µᵥ = 1.65, µᵣ = 1.61, 

⍵ =0.07. Let the refractive index of the material for the yellow light = µᵧ, So

⍵ = (µᵥ - µᵣ)/(µᵧ -1)

→0.07 = (1.65 - 1.61)/(µᵧ - 1)

→µᵧ - 1 = 0.04/0.07

→µᵧ = 1 + 4/7 = 11/7

The minimum deviation in a thin prism

ẟᵧ = (µᵧ - 1)A,     {Where A =angle of the prism}  

→4° = (11/7 - 1)A

→A = 4°/(4/7) = 7°



 

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.                    


ANSWER:  Given, ẟᵣ = 38.4°, ẟᵧ = 38.7° and ẟᵥ = 39.2°, refracting angle of the prism A = 60°.

Since ẟ =(µ-1)A 

→µ - 1 =ẟ/A

Hence, µᵥ - µᵣ = (µᵥ-1) - (µᵣ-1) =ẟᵥ/A - ẟᵣ/A 

                     =(ẟᵥ-ẟᵣ)/A  

and µᵧ - 1 =ẟᵧ/A

Thus the dispersive power of the medium

⍵ = (ẟᵥ - ẟᵣ)/ẟᵧ =(39.2° - 38.4°)/38.7°=0.8°/38.7°

→⍵ = 0.0206   

   


 

 7. Two prisms of identical geometrical shape are combined with their refracting angles oppositely directed. The materials of the prisms have refractive indices1.52 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 prism?                     


ANSWER:  Let the angle of the prism = A.

µᵥ = 1.62 and µ'ᵥ = 1.52

ẟᵥ = (µᵥ -1)A and ẟ'ᵥ =(µ'ᵥ -1)A
Diagram for Q-7

For the oppositely combined prisms, the deviation

ẟ = ẟᵥ - ẟ'ᵥ = (µᵥ-1)A - (µ'ᵥ-1)A, {Given ẟ = 1.0°}

→ẟ = (µᵥ-µ'ᵥ)A 

→1.0° =(1.62-1.52)A

→A = 1.0°/0.10 =10°    

   


 

8. Three thin prisms are combined as shown in the figure (20-E1). The refractive indices of the crown glass for red, yellow and violet rays are µᵣ, µᵧ and µᵥ respectively and those for the flint glass are µᵣ', µᵧ' and µᵥ' 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. 
The figure for Q-8
              


ANSWER:  (a) The average deviation for the single crown glass prism

ẟᵧ = (µᵧ-1)A, and for the Flint glass prism

ẟ'ᵧ = (µ'ᵧ-1)A' 

For the given combination, the net average deviation will be =2ẟᵧ - ẟ'ᵧ 

For no net angular deviation

2ẟᵧ - ẟ'ᵧ = 0

→2ẟᵧ = ẟ'ᵧ

→2(µᵧ -1)A = (µ'ᵧ -1)A' 

→A'/A = 2(µᵧ-1)/(µ'ᵧ-1)

 

(b) The angular dispersion of the single Crown glass prism
ẟᵥ - ẟᵣ = (µᵥ-µᵣ)A,   and the angular dispersion of the single Flint glass prism is
ẟ'ᵥ - ẟ'ᵣ = (µ'ᵥ-µ'ᵣ)A'
The net angular dispersion of the given combination will be
=2(µᵥ-µᵣ)A - (µ'ᵥ-µ'ᵣ)A' 
For no net angular dispersion

2(µᵥ-µᵣ)A - (µ'ᵥ-µ'ᵣ)A' = 0
→2(µᵥ-µᵣ)A = (µ'ᵥ-µ'ᵣ)A'
→A'/A = 2(µᵥ-µᵣ)/(µ'ᵥ-µ'ᵣ)  

9. A thin prism of crown glass (µᵣ = 1.515, µᵥ = 1.525) and a thin prism of flint glass (µᵣ = 1.612, µᵥ =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.               


ANSWER: The angular dispersion produced by the Crown glass prism = ẟᵥ-ẟᵣ =(µᵥ-µᵣ)A

= (1.525-1.515)*5° =0.010*5° =0.05°

Similarly, the angular dispersion produced by the Flint glass prism = (1.632-1.612)*5° =0.020*5° =0.10°.

Since the prisms are similarly directed in the combination, the angular dispersions will add on.

Hence the angular dispersion produced by the combination =0.05°+0.10° =0.15°             

   

 


10. A thin prism of angle 6.0°, ⍵ = 0.07 and µᵧ = 1.50 is combined with another thin prism having ⍵ = 0.08 and µᵧ = 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).               


ANSWER: Given that, for the first prism, A = 6.0°, ⍵ =0.07, µᵧ =1.50

⍵ = (µᵥ-µᵣ)/(µᵧ-1)

→0.07 = (µᵥ-µᵣ)/(1.50-1)

→(µᵥ -µᵣ) = 0.07*0.50 =0.035

Similarly, for the second prism
(µᵥ-µᵣ) =⍵(µᵧ-1) =0.08*(1.60-1) =0.08*0.60 =0.048

(a) Let the angle of the second prism = A'.

The average deviation of the mean ray of the first prism ẟ=(µᵧ-1)A

and of the second prism ẟ' =(µ'ᵧ-1)A'

Since the combination produces no deviation,

ẟ = ẟ'

→(µᵧ-1)A = (µ'ᵧ-1)A'

→(1.50-1)*6° = (1.60-1)*A'

→0.50*6° = 0.60*A'

→A' = 3.0°/0.6 =30°/6 =5°

           

(b) The angular dispersion of the first prism

=(µᵥ-µᵣ)A

=(0.035)*6°

=0.21° 

And the angular dispersion of the second prism

=(µᵥ-µᵣ)A' 

=(0.048)*5°

=0.24°  

Hence the net angular dispersion =0.24°-0.21° =0.03°.


(c) If the prisms are similarly directed, the net deviation in the mean ray = ẟ + ẟ'

=(µᵧ-1)A + (µ'ᵧ-1)A'

=(1.50-1)*6° + (1.60-1)*5°

=0.50*6° + 0.60*5°

=3.0° + 3.0°

=6.0°


(d) The angular dispersion of the combination in the case (c) will be the sum of each prism.
i.e. = 0.24° +0.21° = 0.45°


 

11. The refractive index of a material M1 changes by 0.014 and that of another material M2 changes by 0.024 as the color of the light is changed from red to violet. Two thin prisms are 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.         


ANSWER:  (a) For M1, µᵥ-µᵣ=0.014, A =5.3°, Hence the angular dispersion of sine prism =(µᵥ-µᵣ)A

=0.014*5.3°

=0.0742°

For M2, µᵥ-µᵣ = 0.024, A =3.7°

Hence angular dispersion of single prism of this material =0.024*3.7° =0.0888°               

Since the prisms are oppositely directed, the angular dispersion produced by the combination

=0.0888°-0.0742°

=0.0146°

   

(b) When the prisms are similarly directed, the angular dispersion produced by the combination
=0.0888°+0.0742°
=0.163°

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