Q.A prism is made of glass of unknown refractive index. A parallel beam of light is incident on a face of the prism. The angle of minimum deviation is measured to be $40^\circ$. What is the refractive index of the material of the prism? The refracting angle of the prism is $60^\circ$. If the prism is placed in water (refractive index $1.33$), predict the new angle of minimum deviation of a parallel beam of light.

Prism formula: $n=\frac{\sin[(A+\delta_m)/2]}{\sin(A/2)}$, $A=60^\circ$, $\delta_m=40^\circ$.

In water, the relative index is $n_{rel}=n_g/n_w=1.532/1.33\approx1.15$. Re-applying the prism formula: …

Step 1 — refractive index of the glass in air

Step 2 — relative refractive index in water

The prism formula's $n$ is always the refractive index of the prism relative to the surrounding medium. In water, that relative index is

Step 3 — new angle of minimum deviation

Using the prism formula again, now with $n_{rel}$ in place of $n_g$, and the same prism angle $A=60^\circ$:

$n_{rel} = \frac{\sin\left(\frac{A+\delta_m'}{2}\right)}{\sin(A/2)} \;\Rightarrow\; \sin\left(\frac{A+\delta_m'}{2}\right) = 1.152 \times 0.5 = 0.576$ …

Method: Prism Formula (Minimum Deviation)

This is a direct application of the Prism Formula under minimum deviation condition.

Step 1: Recall the Prism Formula

For a prism at minimum deviation:

Where:

• $n$ = refractive index of prism material (relative to surrounding medium)
• $A$ = refracting angle of prism ($60^\circ$)
• $D_m$ = angle of minimum deviation

Step 2: Find refractive index in air

Given: $A = 60^\circ$, $D_m = 40^\circ$

So the refractive index of glass is 1.532.

Step 3: For prism in water — use relative refractive index

When the prism is in water, the effective refractive index is:

Step 4: Apply prism formula again for new $D_m'$

Using the same prism formula with $n_{\text{relative}}$: …

Here are the common mistakes students make on this classic prism problem, along with how to avoid each one.

1. Forgetting the Formula for Refractive Index

Mistake:

Students often try to guess or derive the refractive index without using the standard formula:

They might incorrectly use $\mu = \frac{\sin i}{\sin r}$ directly without linking it to the prism geometry.

How to avoid:

• Memorise the prism formula as a standard result.
• Always write it down first before plugging numbers.
• Remember: $A$ is the refracting angle (given as $60^\circ$), and $\delta_m$ is the angle of minimum deviation ($40^\circ$).

2. Incorrect Substitution of Angles

Mistake:

Students sometimes substitute $A = 60^\circ$ and $\delta_m = 40^\circ$ directly into the formula without computing the half-angles correctly.

Example of error:

Writing $\sin(60^\circ + 40^\circ)$ instead of $\sin\left(\frac{60^\circ + 40^\circ}{2}\right)$.

How to avoid:

• Always compute the half-sum first:

• Then compute $\sin 50^\circ$ and $\sin 30^\circ$ separately.

3. Using Degrees in Calculator Without Checking Mode

Mistake:

Many students forget to set their calculator to degree mode when evaluating $\sin 50^\circ$ and $\sin 30^\circ$, leading to wildly wrong answers.

How to avoid:

• Always check the calculator mode before starting.
• If you get a refractive index like $0.5$ or $2.5$, you’ve likely used radians by mistake.

4. Forgetting to Change the Medium for the Second Part

Mistake:

When the prism is placed in water, students often use the same formula with the same $\mu$ for glass, ignoring that the relative refractive index changes.

How to avoid:

• The refractive index of glass relative to water is:

• Use this relative index in the prism formula to find the new $\delta_m$.

5. Assuming the Angle of Minimum Deviation Stays the Same

Mistake:

Some students think $\delta_m$ is a property of the prism alone and doesn’t change with the surrounding medium.

How to avoid:

• Understand that $\delta_m$ depends on the relative refractive index between prism and surrounding medium.
• When the surrounding medium changes, the bending of light changes, so $\delta_m$ changes.

6. Arithmetic Errors in the Second Part

Mistake: …