Prism Refraction and Deviation

1. Visualizing the Path of Light Through a Prism

Before we get into the mathematics, let's build a strong mental model of what happens when a light ray encounters a prism. A prism is simply a transparent optical element with flat, polished surfaces that are inclined at an angle to each other. This inclination is what distinguishes it from a glass slab, where the faces are parallel.

To see this in action, let's start with a short video that demonstrates how you would physically trace a ray of light through a prism in a lab.

Now that you've seen the experimental process, let's consolidate our understanding with a clear, labeled diagram.

• Incident Ray (PE): The ray of light that strikes the first surface of the prism.
• Refracted Ray (EF): The ray of light as it travels inside the prism.
• Emergent Ray (FS): The ray of light that leaves the second surface of the prism.
• Angle of the Prism (A): The angle between the two refracting surfaces (faces AB and AC).
• Angle of Incidence (i): The angle between the incident ray and the normal at the point of incidence.
• Angle of Refraction (r₁): The angle between the refracted ray and the normal at the first surface.
• Angle of Incidence on the second face (r₂): The angle between the refracted ray and the normal at the second surface.
• Angle of Emergence (e): The angle between the emergent ray and the normal at the point of emergence.
• Angle of Deviation (δ or D): The angle between the extended incident ray and the emergent ray. This measures the total "bend" in the light's path caused by the prism.

Notice a key observation from both the video and the diagram: The emergent ray bends towards the base of the prism (BC).

2. The Geometry of Deviation: Key Relationships

The angles we've just defined aren't independent; they are linked by the geometry of the prism. Deriving these relationships is a fundamental skill. We will now watch a video that walks through the geometric proofs for two essential formulas.

Let's summarize those crucial derivations and results.

Summary of Key Formulas

By analyzing the geometry of the ray path, we arrive at two indispensable equations:

1. Relation for the Prism Angle:
   In the quadrilateral AQNR (using the diagram from the video), the sum of angles is 360°. Since the normals make 90° angles with the prism faces, we can show that:
   $A+\angle QNR=180^{\circ }$
   In the triangle QNR, we have $r_1+r_2+\angle QNR=180^{\circ }$.
   Comparing these two gives us our first key formula:
   $A=r_1+r_2$

2. Relation for the Angle of Deviation:
   The total deviation $\delta$ is the sum of the deviation at the first surface (${\delta }_1=i_1-r_1$) and the deviation at the second surface (${\delta }_2=i_2-r_2$).
   $\delta ={\delta }_1+{\delta }_2=(i_1-r_1)+(i_2-r_2)$
   Rearranging this, we get:
   $\delta =(i_1+i_2)-(r_1+r_2)$
   Substituting our first formula ($A=r_1+r_2$) into this gives our second key formula:
   $\delta =i_1+i_2-A$

These two equations, combined with Snell's law at each surface, are the complete toolkit you need to analyze the path of a ray through any prism.

3. Applying the Concepts: A Worked Example

Let's put these formulas into practice. This problem integrates Snell's Law with the geometric relations we just derived.

Problem: A ray of light is incident at an angle of 60° on one face of an equilateral prism ($A=60^{\circ }$). The refractive index of the prism material is $\sqrt{3}$. Calculate:
a) The angle of refraction at the first face ($r_1$)
b) The angle of incidence at the second face ($r_2$)
c) The angle of emergence ($e$)
d) The total angle of deviation ($\delta$)

Solution:

Step 1: Apply Snell's Law at the first face (air to prism)
We have $i_1=60^{\circ }$, $n_{air}=1$, and $n_{prism}=\sqrt{3}$.
$n_{air}\sin i_1=n_{prism}\sin r_1$
$1\times \sin 60^{\circ }=\sqrt{3}\times \sin r_1$
$\frac{\sqrt{3}}{2}=\sqrt{3}\sin r_1$
$\sin r_1=\frac{1}{2}⟹{\mathbf{r}}_1=30^{\circ }$

Step 2: Use the prism angle formula to find $r_2$
We know $A=60^{\circ }$ and we just found $r_1=30^{\circ }$.
$A=r_1+r_2$
$60^{\circ }=30^{\circ }+r_2⟹{\mathbf{r}}_2=30^{\circ }$

Step 3: Apply Snell's Law at the second face (prism to air)
Now, the light goes from the prism back into the air. The angle of incidence is $r_2=30^{\circ }$.
$n_{prism}\sin r_2=n_{air}\sin e$
$\sqrt{3}\times \sin 30^{\circ }=1\times \sin e$
$\sqrt{3}\times \frac{1}{2}=\sin e$
$\sin e=\frac{\sqrt{3}}{2}⟹\mathbf{e}=60^{\circ }$

Step 4: Use the deviation formula to find $\delta$
We have all the necessary angles: $i_1=60^{\circ }$, $e=60^{\circ }$, and $A=60^{\circ }$.
$\delta =i_1+e-A$
$\delta =60^{\circ }+60^{\circ }-60^{\circ }⟹\mathbf{\delta }=60^{\circ }$

Conclusion

In this lesson, we established the fundamental principles of how light travels through a prism. By mastering the visual path and the underlying geometry, you've built a solid foundation for all future work on this topic.

Key Takeaways:

• A light ray passing through a prism is refracted at both surfaces, causing it to deviate from its original path, bending towards the prism's base.
• The geometry of this path is described by two essential formulas:
  • $A=r_1+r_2$
  • $\delta =i_1+i_2-A$ (where $i_2$ is often written as $e$)
• To solve any prism problem, you need to systematically apply these two formulas along with Snell's law at each of the two interfaces.

Preview of the Next Lesson:
In our worked example and the mini-exercise, you saw scenarios where the path of the light was symmetric ($i=e$). This is a very special and important case. In our next lesson, we will investigate this condition, known as minimum deviation. We will derive the famous prism formula, $\mu =\frac{\sin ((A+{\delta }_m)/2)}{\sin (A/2)}$, which is a powerful tool for finding the refractive index of a material and a favorite topic for NEET questions.