Question

A ray of light is incident at angle of $45^\circ$ on one face of a prism with an equilateral triangular base. If it passes symmetrically through the prism, find the angle of minimum deviation for the prism

Hint:

The keyword "symmetrically" in prism problems is a massive hint. It instantly tells you two things: $i = e$ and the deviation is at its absolute minimum.

Answer 1

Concept:
• When a ray of light passes "symmetrically" through a prism, the angle of incidence ($i$) is exactly equal to the angle of emergence ($e$).
• This symmetric path occurs only when the prism is in the state of minimum deviation.
• The general prism formula relating incidence, emergence, prism angle ($A$), and deviation ($\delta$) is: $i + e = A + \delta$.

Step 1: Identify the given parameters
The prism has an equilateral triangular base, which means all its internal angles are equal. Therefore, the angle of the prism, $A = 60^\circ$. The angle of incidence is given as $i = 45^\circ$. Because the light passes symmetrically, we know the deviation is at its minimum ($\delta_m$), and the angle of emergence equals the angle of incidence: $e = i = 45^\circ$.

Step 2: Apply the minimum deviation formula
Use the standard prism equation: \[ i + e = A + \delta_m \]
Substitute the known values ($e = i$): \[ i + i = A + \delta_m \]
\[ 2i = A + \delta_m \]
\[ 2(45^\circ) = 60^\circ + \delta_m \]
\[ 90^\circ = 60^\circ + \delta_m \]

Step 3: Solve for $\delta_m$
Rearrange to find the angle of minimum deviation: \[ \delta_m = 90^\circ - 60^\circ \]
\[ \delta_m = 30^\circ \]

Step 4: Conclusion
The angle of minimum deviation for the equilateral prism is $30^\circ$.