Question

Derive the relation for the refractive index ($\mu$) of a prism in terms of angle of minimum derivation ($\delta_m$) and angle of prism (A).
Acute{}

Hint:

The condition for minimum deviation ($i=e$) fundamentally implies that the light ray traveling inside the prism is perfectly parallel to the base of the prism (assuming the prism is an isosceles or equilateral triangle).

Answer 1

Concept:
• As light sequentially traverses the two slanted interfaces of a glass prism, it undergoes consecutive refractions, causing an overall angular deviation.
• The overall angle of deviation intricately depends on the exact angle of incidence.
• There is exactly one unique angle of incidence where this total deviation hits its absolute minimum value, creating high geometric symmetry.

Step 1: Establish the basic geometric prism equations
For an incoming light ray undergoing refraction through a solid prism, standard optical geometry dictates two fundamental relationships:
First, the refracting angle of the prism ($A$) is equal to the sum of the two internal angles of refraction ($r_1$ and $r_2$):
\[ A = r_1 + r_2 \quad \text{--- (Equation 1)} \]
Second, the total angle of deviation ($\delta$) produced by the prism is strictly related to the angle of incidence ($i$), angle of emergence ($e$), and the prism angle:
\[ \delta = i + e - A \quad \text{--- (Equation 2)} \]

Step 2: Apply the specific condition for minimum deviation
Experimental observation proves that the deviation angle $\delta$ rigorously reaches its absolute minimum value ($\delta_m$) exactly when the light ray travels perfectly symmetrically through the prism.
Under this highly symmetric condition:
The angle of incidence is exactly equal to the angle of emergence ($i = e$).
The internal angles of refraction at both faces are exactly equal ($r_1 = r_2 = r$).

Step 3: Solve for $r$ and $i$ in terms of $A$ and $\delta_m$
Substitute the symmetric condition $r_1 = r_2 = r$ directly into Equation 1:
\[ A = r + r = 2r \]
\[ r = \frac{A}{2} \quad \text{--- (Equation 3)} \]
Next, seamlessly substitute the conditions $\delta = \delta_m$ and $i = e$ strictly into Equation 2:
\[ \delta_m = i + i - A \]
\[ \delta_m = 2i - A \]
Rearrange to explicitly solve for the angle of incidence $i$:
\[ 2i = A + \delta_m \]
\[ i = \frac{A + \delta_m}{2} \quad \text{--- (Equation 4)} \]

Step 4: Apply Snell's Law to derive the final formula
According to Snell's law applied at the very first interface, the refractive index ($\mu$) of the prism material relative to the surrounding air is given by:
\[ \mu = \frac{\sin i}{\sin r_1} \]
Since $r_1 = r$ during minimum deviation, substitute Equations 3 and 4 directly into this law:
\[ \mu = \frac{\sin \left( \frac{A + \delta_m}{2} \right)}{\sin \left( \frac{A}{2} \right)} \]
This gives the required expression for the refractive index accurately in terms of the measurable angles.