The refractive indices of the crown glass for blue and red light are 1.51 and 1.49 respectively and

Q: Why do we use the mean refractive index for the no-deviation condition?

White light consists of a spectrum of colors. When we say "no deviation of the incident light," we refer to the mean ray, which is typically the yellow light. The mean refractive index represents this average behavior.

Q: What does the negative sign for the prism angle indicate?

A negative prism angle means that the second prism is inverted relative to the first one. Their refracting angles point in opposite directions, allowing their deviations to cancel each other out.

Q: Can we have dispersion without deviation?

Yes! By choosing two different materials (like crown and flint glass) with different dispersive powers, we can combine them such that their mean deviations cancel out, but a net dispersion still remains. This principle is used in direct-vision spectroscopes.

Q: Why is the net dispersion negative?

The negative sign indicates that the order of colors in the dispersed spectrum is reversed compared to what a single upright prism would produce. The flint glass prism dominates the dispersion but in the opposite direction.

Q: What is the difference between deviation and dispersion?

Deviation is the overall bending of a light ray from its original path. Dispersion is the splitting of white light into its constituent colors due to different refractive indices for different wavelengths.

Question

The refractive indices of the crown glass for blue and red light are 1.51 and 1.49 respectively and those of the flint glass are 1.77 and 1.73 respectively. An isosceles prism of angle $6^\circ$ is made of crown glass. A beam of white light is incident at a small angle on this prism. The other flint glass isosceles prism is combined with the crown glass prism such that there is no deviation of the incident light.
(a) Determine the angle of the flint glass prism.
(b) Calculate the net dispersion of the combined system.

Sigma Percentile · Accepted Answer

The phenomenon of **dispersion without deviation** is a fascinating application of optics, commonly used in direct-vision spectroscopes. It allows us to split white light into its constituent colors without changing the overall direction of the beam. Let's dive into how this is achieved using a combination of crown and flint glass prisms!

### The Principle of No Deviation

When a beam of white light passes through a prism, it undergoes both deviation (bending) and dispersion (splitting of colors). To achieve dispersion without deviation, we combine two prisms made of different materials—in this case, crown glass and flint glass.

The condition for no net deviation is that the mean deviation produced by the first prism must be exactly canceled by the mean deviation produced by the second prism. For a thin prism, the deviation $\delta$ is given by:
$$ \delta = (\mu - 1)A $$
where $\mu$ is the mean refractive index and $A$ is the angle of the prism.

For the combination to produce zero net deviation, the sum of their individual deviations must be zero:
$$ (\mu_1 - 1)A_1 + (\mu_2 - 1)A_2 = 0 $$

### Calculating the Mean Refractive Indices

The mean refractive index is the average of the refractive indices for the extreme colors (blue and red).

For the crown glass prism:
$$ \mu_1 = \frac{\mu_{b1} + \mu_{r1}}{2} = \frac{1.51 + 1.49}{2} = 1.50 $$

For the flint glass prism:
$$ \mu_2 = \frac{\mu_{b2} + \mu_{r2}}{2} = \frac{1.77 + 1.73}{2} = 1.75 $$

### Finding the Angle of the Flint Glass Prism

We are given that the angle of the crown glass prism is $A_1 = 6^\circ$. Substituting our values into the no-deviation condition:
$$ (1.50 - 1)(6^\circ) + (1.75 - 1)A_2 = 0 $$
$$ 0.5 	imes 6^\circ + 0.75 A_2 = 0 $$
$$ 3^\circ + 0.75 A_2 = 0 $$
$$ A_2 = -\frac{3^\circ}{0.75} = -4^\circ $$

The negative sign is crucial here! It indicates that the flint glass prism must be placed **inverted** relative to the crown glass prism. So, the angle of the flint glass prism is $4^\circ$.

### Calculating the Net Dispersion

Now, let's find the net dispersion produced by this combination. Dispersion $	heta$ is the difference in deviation between the blue and red rays:
$$ 	heta = (\mu_b - \mu_r)A $$

The net dispersion is the algebraic sum of the dispersions from both prisms:
$$ 	ext{Net Dispersion} = 	heta_1 + 	heta_2 $$
$$ 	ext{Net Dispersion} = (\mu_{b1} - \mu_{r1})A_1 + (\mu_{b2} - \mu_{r2})A_2 $$

Substituting the given values:
$$ 	ext{Net Dispersion} = (1.51 - 1.49)(6^\circ) + (1.77 - 1.73)(-4^\circ) $$
$$ 	ext{Net Dispersion} = (0.02)(6^\circ) + (0.04)(-4^\circ) $$
$$ 	ext{Net Dispersion} = 0.12^\circ - 0.16^\circ = -0.04^\circ $$

The net dispersion is **$-0.04^\circ$**. The negative sign implies that the final spectrum is reversed compared to the spectrum that would be produced by the crown glass prism alone. The flint glass, having a higher dispersive power, dominates the dispersion process even though it has a smaller prism angle!

Visualized Solution

$Combination of Prisms$

$Condition for No Deviation$

$Mean Refractive Indices$

$Angle of Flint Glass Prism$

$Net Dispersion$

$Calculating Net Dispersion$

$Final Answer$

The Sigma Insight: Refraction and Dispersion through Prism

The Principle of No Deviation

Calculating the Mean Refractive Indices

Finding the Angle of the Flint Glass Prism

Calculating the Net Dispersion

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Combination of Prisms

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Condition for No Deviation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Mean Refractive Indices

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Angle of Flint Glass Prism

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Net Dispersion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating Net Dispersion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Answer