Answers to Problems on (Dispersion and Spectra) HC Verma's Questions for Short Answer

 Q#1

The equation ⍵ = (µᵥ – µᵣ)/(µ – 1) was derived for a prism having a small refracting angle. It is also valid for a prism of large refracting angle? Is it also valid for a glass slab or a glass sphere?

Answer:
The given equation is derived assuming the refracting angle A to be small so that

Sin (A+ẟ)/2 ≈ (A+ẟ)/2 and
Sin (A/2) ≈ A/2.

So it is not valid for a prism of large refracting angle.

For a glass sphere, the refracting angle will be large. Hence it is not valid for a glass sphere.

For a glass slab, the refracting angle A = 0. But,

ẟ = (µ - 1)A = 0.

So, this equation for ⍵ cannot be derived for A = 0.

Q#2
Can the dispersive power ⍵ = (µᵥ – µᵣ)/(µ – 1) be negative? What is the sign of ⍵ if a hollow prism is immersed in water?     

Answer:
In the equation ⍵ = (µᵥ – µᵣ)/(µ – 1)

µᵥ is always greater than µᵣ for the transparent prism kept in a medium having a lesser refractive index than it.

So µᵥ – µᵣ > 0. Also µ > 1, so µ - 1 >0.

Hence the dispersive power ⍵ will be positive.
If a hollow prism is immersed in water the equation of the dispersive power will become

⍵ = (1/µᵥ – 1/µᵣ)/(1/µ – 1)

So, both the numerator and the denominator will become negative and we will get ⍵ positive. So, in normal situations ⍵ will not be negative.

Q#3
If three identical prisms are combined, is it possible to pass a beam that emerges undeviated? Undispersed?

Answer:
Since the prisms are identical, they can be combined in two ways - First, having each one's vertex near each other (at the top) and second, two vertices and one base on the top.
The first case will make it a prism with the refracting angle three times larger. So, there will be a deviation as well as dispersion.

The second case will make it a similar prism to one of the identical prisms with the same refracting angle. So, in this case also there will be a deviation and the dispersion. Hence the answer is "No".

Q#4
"Monochromatic light should be used to produce pure spectrum". Comment on this statement. 

Answer:
The spectrum is obtained on a white wall or paper when different wavelengths of light get deviated through different angles by a dispersing element. When no two colors overlap each other in this spectrum, it is called a pure spectrum. If monochromatic light is used there will be only one color, not a spectrum.

Q#5
Does the focal length of a lens depend on the color of the light used? Does the focal length of a mirror depend on the color?

Answer:
The focal length of a lens is given by the lens maker's formula

1/f = (µ – 1){1/R₁ – 1/R₂}

Clearly, the focal length increases as µ decreases. µ depends on the color of the light. So the focal length of a lens depends on the color of the light.

Since there is no refraction in the mirror and it works on the laws of reflection which is the same for all colors. For example, for the same angle of incidence, the angle of reflection will be the same for all colors. Hence the focal length of a mirror does not depend on the color of the light.  

Q#6
Suggest a method to produce a rainbow in your house.

Answer:
The principle behind making a rainbow is to disperse the white light. One method to make a rainbow inside a room is to allow a narrow beam of light to fall on a surface of the water in a glass. For this sunray is allowed to make a narrow beam from a partly closed window. Transparent glass is filled with clear water and a paper with a slit is glued to the upper portion of the glass. The glass is placed such that the sun rays fall on the slit and the emerging light is allowed to fall on a white paper. We get a rainbow on the paper. See the diagram below.

Make sure that the glass of water is very full. Place the glass on a white floor or white piece of paper, making sure the sun’s rays are shining through the slot in your paper and hitting the surface of the water.
You should see a mini-rainbow appear below the glass!

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