Angular Dispersion and Dispersive Power of Prisms

1. Angular Dispersion: Measuring the Spread

The first and most direct way to quantify dispersion is to measure the angular separation between the two extreme colors of the spectrum: violet and red. This separation is called angular dispersion.

Definition: Angular dispersion ($\theta$) is the difference between the angle of deviation for violet light (${\delta }_v$) and the angle of deviation for red light (${\delta }_r$).

$\theta ={\delta }_v-{\delta }_r$

For a thin prism (where the prism angle $A$ is small, typically $\le 10^{\circ }$), we can easily derive a more practical formula. Recall the thin prism deviation formula from our previous lessons:
$\delta \approx (\mu -1)A$

Applying this to violet and red light separately:

• For violet light: ${\delta }_v=({\mu }_v-1)A$
• For red light: ${\delta }_r=({\mu }_r-1)A$

Now, substituting these into the definition of angular dispersion:
$\theta =({\mu }_v-1)A-({\mu }_r-1)A$
$\theta =({\mu }_{v}A-A)-({\mu }_{r}A-A)$
$\theta ={\mu }_{v}A-{\mu }_{r}A$

This gives us the formula for angular dispersion for a thin prism:
$\theta =({\mu }_v-{\mu }_r)A$

Notice that angular dispersion depends on two factors:

1. The material's properties (${\mu }_v-{\mu }_r$).
2. The prism's angle ($A$).

To see this derivation explained visually, the following video is very helpful.

2. Dispersive Power: An Intrinsic Material Property

While angular dispersion tells us the total spread, it doesn't tell the whole story. A prism could produce a large angular dispersion simply by having a large angle $A$. We need a way to describe the material's inherent ability to disperse light, independent of the prism's geometry. This is where dispersive power comes in.

Definition: Dispersive power ($\omega$) is the ratio of the angular dispersion ($\theta$) to the deviation suffered by the mean (average) color, which is typically taken as yellow light (${\delta }_y$).

$\omega =\frac{\text{Angular Dispersion}}{\text{Mean Deviation}}=\frac{\theta }{{\delta }_y}$

Dispersive power is a dimensionless quantity that tells us how much a material spreads out colors relative to how much it bends them on average. A high dispersive power means a large spread for a given average bend.

Now, let's derive the formula in terms of refractive indices for a thin prism:

• Angular Dispersion: $\theta =({\mu }_v-{\mu }_r)A$
• Mean Deviation: ${\delta }_y=({\mu }_y-1)A$

Substituting these into the definition:
$\omega =\frac{({\mu }_v-{\mu }_r)A}{({\mu }_y-1)A}$

The prism angle $A$ cancels out, leaving us with the standard formula for dispersive power:
$\omega =\frac{{\mu }_v-{\mu }_r}{{\mu }_y-1}$

This is a crucial result. It shows that dispersive power depends only on the refractive indices of the material, not on the shape of the prism.

A Practical Tip for NEET Problems

Often, a problem will give you ${\mu }_v$ and ${\mu }_r$ but not ${\mu }_y$. In such cases, since yellow light is roughly in the middle of the visible spectrum, you can approximate its refractive index as the average of the two extremes:
${\mu }_y\approx \frac{{\mu }_v+{\mu }_r}{2}$

This approximation is frequently needed to solve exam questions.

The following video provides an excellent summary of dispersive power, including its derivation and how to apply the formulas in a problem-solving context.

3. Application and Problem-Solving

Let's solidify these concepts with a couple of typical problems. Your goal in the exam is to quickly identify the given quantities and choose the correct formula.

Example 1: Calculating Dispersive Power from Refractive Indices

This problem is a direct application of the formula.

• Given: ${\mu }_r=1.613$, ${\mu }_y=1.620$, ${\mu }_v=1.632$.
• Calculation:
  $\omega =\frac{1.632-1.613}{1.620-1}=\frac{0.019}{0.620}\approx 0.0306$

Example 2: Calculating Dispersive Power without ${\mu }_y$

This example forces us to use the approximation for the mean refractive index.

• Given: ${\mu }_v=1.6$, ${\mu }_r=1.4$.
• Step 1: Find ${\mu }_y$.
  ${\mu }_y=\frac{{\mu }_v+{\mu }_r}{2}=\frac{1.6+1.4}{2}=1.5$
• Step 2: Calculate $\omega$.
  $\omega =\frac{{\mu }_v-{\mu }_r}{{\mu }_y-1}=\frac{1.6-1.4}{1.5-1}=\frac{0.2}{0.5}=0.4$

Mastering these two types of calculations will equip you well for questions on this topic.

Conclusion

Today we moved from a qualitative understanding of dispersion to a quantitative one. This is a critical step for your exam preparation, where applying the correct formula is key.

Key Takeaways:

• Angular Dispersion ($\theta$) measures the angular spread of the spectrum. For a thin prism, $\theta =({\mu }_v-{\mu }_r)A$. It depends on both the material and the prism's angle.
• Dispersive Power ($\omega$) measures the material's intrinsic ability to disperse light. The formula is $\omega =\frac{{\mu }_v-{\mu }_r}{{\mu }_y-1}$. It is a dimensionless property of the material only.
• If ${\mu }_y$ is not given, it can be approximated as the average: ${\mu }_y\approx ({\mu }_v+{\mu }_r)/2$.

Preview of the Next Lesson:
You now have all the necessary theoretical tools related to prisms: the prism formula, the concept of dispersion, and the formulas for angular dispersion and dispersive power. In our next lesson, we will dedicate the entire session to solving a variety of NEET-style problems involving prisms and dispersion. This will involve single prisms, combinations of prisms, and scenarios that test your conceptual and mathematical understanding.