Prism Formula and Minimum Deviation

1. The Concept of Minimum Deviation

If you were to conduct an experiment where you vary the angle of incidence ($i$) for a prism and measure the corresponding angle of deviation ($\delta$), you would find a distinct pattern. A graph of $\delta$ versus $i$ shows that the deviation angle initially decreases as you increase the angle of incidence, reaches a single minimum value, and then begins to increase again.

This lowest possible value of the deviation angle is called the angle of minimum deviation (${\delta }_m$). It occurs at a unique angle of incidence where the light ray passes symmetrically through the prism.

This symmetry implies two crucial conditions:

• The angle of incidence is equal to the angle of emergence: $i=e$.
• The angles of refraction inside the prism are equal: $r_1=r_2$.

Let's watch a video that explains this concept clearly using the $\delta$ vs. $i$ graph.

2. Deriving the Prism Formula

Now that we have the conditions for minimum deviation, we can combine them with the geometric formulas from our last lesson to derive the famous prism formula. This is a classic derivation that connects the macroscopic properties of the prism ($A$, ${\delta }_m$) to the microscopic property of its material ($\mu$).

Let's start with our known equations:

• General deviation: $\delta =i+e-A$
• Prism angle: $A=r_1+r_2$
• Snell's Law at the first face: $1\cdot \sin i=\mu \sin r_1$

At minimum deviation, we apply our new conditions:

• $\delta$ becomes ${\delta }_m$
• $i=e$
• $r_1=r_2=r$ (let's just call it $r$)

Now, let's substitute these into the first two equations:

1. Finding $i$ in terms of ${\delta }_m$ and $A$:
   ${\delta }_m=i+i-A⟹{\delta }_m=2i-A$
   Rearranging this to solve for $i$, we get:
   $i=\frac{A+{\delta }_m}{2}$

2. Finding $r$ in terms of $A$:
   $A=r+r⟹A=2r$
   Rearranging this to solve for $r$, we get:
   $r=\frac{A}{2}$

We now have expressions for the angle of incidence $i$ and the angle of refraction $r$ under the specific condition of minimum deviation. The final step is to substitute these into Snell's Law.

To see this derivation process clearly, let's watch the next part of the Khan Academy video.

The result of this derivation is the prism formula:

$\mu =\frac{\sin \left(\frac{A+{\delta }_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$

This equation is extremely powerful. It allows us to determine the refractive index of a material by simply measuring the prism's angle and the angle of minimum deviation, which are both macroscopic, measurable quantities.

3. Applying the Prism Formula: NEET-Style Problems

As a NEET aspirant, your primary goal is to apply these formulas quickly and accurately. Let's work through a typical problem.

Problem: The angle of minimum deviation for an equilateral prism is 30°. Find the speed of light inside the prism. (Use speed of light in vacuum, $c=3\times 10^8$ m/s).

Solution Strategy:

1. Identify the given information.
2. Use the prism formula to calculate the refractive index ($\mu$).
3. Use the definition of refractive index ($\mu =c/v$) to find the speed of light in the prism ($v$).

Step-by-step Solution:

1. Given:

   • Equilateral prism $⟹A=60^{\circ }$.
   • Angle of minimum deviation ${\delta }_m=30^{\circ }$.

2. Calculate $\mu$ using the prism formula:
   $\mu =\frac{\sin \left(\frac{A+{\delta }_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$
   $\mu =\frac{\sin \left(\frac{60^{\circ }+30^{\circ }}{2}\right)}{\sin \left(\frac{60^{\circ }}{2}\right)}=\frac{\sin \left(45^{\circ }\right)}{\sin \left(30^{\circ }\right)}$
   $\mu =\frac{1/\sqrt{2}}{1/2}=\frac{2}{\sqrt{2}}=\sqrt{2}$

3. Calculate the speed of light $v$:
   $\mu =\frac{c}{v}⟹v=\frac{c}{\mu }$
   $v=\frac{3\times 10^8\text{ m/s}}{\sqrt{2}}\approx 2.12\times 10^8\text{ m/s}$

The speed of light inside the prism is approximately $2.12\times 10^8$ m/s.

Now, let's watch a video with more worked examples to solidify your problem-solving skills.

For a quick reference and more practice, you can consult the following resource.

Conclusion

Today we took a deep dive into the important case of minimum deviation. By understanding the condition of symmetry, we were able to derive the prism formula, a crucial tool for both theoretical analysis and practical problem-solving.

Key Takeaways:

• Minimum deviation (${\delta }_m$) is the smallest possible deviation angle for a given prism.
• It occurs when the light ray passes symmetrically, meaning the angle of incidence equals the angle of emergence ($i=e$) and the internal refraction angles are equal ($r_1=r_2$).
• At minimum deviation, the deviation angle is given by ${\delta }_m=2i-A$.
• The prism formula connects the refractive index to the prism angle and minimum deviation: $\mu =\frac{\sin ((A+{\delta }_m)/2)}{\sin (A/2)}$.

Preview of the Next Lesson:
So far, we have only considered monochromatic light (light of a single color). What happens if we shine white light through a prism? You've probably seen the result: a rainbow. In our next lesson, we will explore this phenomenon, known as dispersion, and learn how to quantify it.