Q.For a glass prism ($\mu = \sqrt{3}$) the angle of minimum deviation is equal to the angle of the prism. Find the angle of the prism.

The key idea is Index Matching: at minimum deviation, the ray passes symmetrically through the prism, so $r = A/2$ and $i = (A + \delta_m)/2$. Here $\delta_m = A$ is given.

Step 1: Snell’s law at the first face:

$\mu = \frac{\sin i}{\sin r}$.

Step 2: Substitute $r = A/2$ and $i = (A + A)/2 = A$:

$\sqrt{3} = \frac{\sin A}{\sin (A/2)}$.

Step 3: Use $\sin A = 2 \sin(A/2) \cos(A/2)$: …

The problem gives you a neat condition: the angle of minimum deviation equals the prism angle itself. That is, $D_m = A$. This is a special case that collapses the standard prism formula into something much simpler.

The key idea is index matching — you have one equation (the refractive index formula for minimum deviation) and one unknown ($A$). The condition $D_m = A$ lets you substitute directly, turning a two-variable problem into a single-variable trigonometric equation.

Let’s walk through it.

1. Recall the standard formula for refractive index $\mu$ of a prism at minimum deviation:

This is derived from Snell’s law and the geometry of symmetric passage through the prism. It’s valid only when the deviation is minimum — which is exactly the case here.

2. Apply the given condition: $D_m = A$. Substitute this into the formula:

3. Simplify using a double-angle identity. Recall $\sin A = 2\sin\frac{A}{2}\cos\frac{A}{2}$. So:

The $\sin\frac{A}{2}$ cancels (provided $A \neq 0$, which is true for a prism).

4. Plug in the given value: $\mu = \sqrt{3}$. So:

5. Solve for $A$. We know $\cos\theta = \frac{\sqrt{3}}{2}$ when $\theta = 30^\circ$ (or $\pi/6$ radians). So: …

Method: Solving Prism Problems Using the Minimum-Deviation Formula

Use this method for any prism question that gives (or asks for) the refractive index, the prism angle $A$, or the minimum deviation $D_m$, when the ray is known to pass through the prism symmetrically.

Steps

Step 1: Write down the minimum-deviation formula

At minimum deviation, the ray inside the prism travels parallel to the base and passes through symmetrically, so the standard relation connecting refractive index $\mu$, prism angle $A$, and minimum deviation $D_m$ is

This is the single formula almost every "prism at minimum deviation" problem reduces to.

Step 2: Substitute whatever relationship the problem gives between $A$ and $D_m$

Some problems give $D_m$ directly as a number; others (like this one) give a relationship between $D_m$ and $A$ (e.g. $D_m = A$, or $D_m = A/2$). Substitute that relationship into the formula so everything is in terms of a single unknown.

Step 3: Simplify using a trigonometric identity …