Minimum deviation for an equilateral prism is 30°, refractive index is:
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For an equilateral prism the angle of the prism is always $A = 60^\circ$, so start from that number before doing anything else. At minimum deviation the ray path inside the prism is symmetric, which means the angle of refraction at each face is exactly half the prism angle, not half the angle of deviation. Fix the refraction angle first using this fact, then bring in the given minimum deviation value to complete the refractive index formula.
Step 1: Understanding the Concept:
An equilateral prism has an angle of prism \( A = 60^\circ \). The refractive index \( \mu \) of the material of the prism is related to the angle of minimum deviation \( \delta_m \) and the angle of the prism \( A \). Step 2: Key Formula or Approach:
The prism formula is:
\[ \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \] Step 3: Detailed Explanation:
1. Given: \( A = 60^\circ \) (equilateral) and \( \delta_m = 30^\circ \).
2. Substitute the values into the formula:
\[ \mu = \frac{\sin\left(\frac{60^\circ + 30^\circ}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)} \]
\[ \mu = \frac{\sin(45^\circ)}{\sin(30^\circ)} \]
3. Using trigonometric values \( \sin 45^\circ = \frac{1}{\sqrt{2}} \) and \( \sin 30^\circ = \frac{1}{2} \):
\[ \mu = \frac{1/\sqrt{2}}{1/2} = \frac{2}{\sqrt{2}} = \sqrt{2} \] Step 4: Final Answer:
The refractive index of the prism is \( \sqrt{2} \).
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Approach Solution -2
Concept:
Rather than plugging straight into the memorised formula, build it from the geometry of a prism at minimum deviation: the ray inside the prism travels symmetrically, so the angle of incidence equals the angle of emergence.
Step 1: Use the prism's angle relation.
For any prism, $A = r_1 + r_2$, where $r_1, r_2$ are the refraction angles at the two surfaces.
Step 2: Apply the minimum-deviation symmetry condition.
At minimum deviation, the ray path is symmetric, so $r_1 = r_2 = r$. This gives $r = \dfrac{A}{2} = \dfrac{60^\circ}{2} = 30^\circ$.
Step 3: Relate the incidence angle to the deviation.
The deviation is $\delta = i_1 + i_2 - A$. At minimum deviation, $i_1 = i_2 = i$, so $\delta_m = 2i - A \Rightarrow i = \dfrac{A+\delta_m}{2} = \dfrac{60^\circ+30^\circ}{2} = 45^\circ$.
Step 4: Apply Snell's law at the first surface.
$\mu = \dfrac{\sin i}{\sin r} = \dfrac{\sin 45^\circ}{\sin 30^\circ} = \dfrac{1/\sqrt{2}}{1/2} = \sqrt{2}$
