When a ray of light passes through a prism and the angle of deviation is minimum, which of the following is true about the angles of incidence and emergence?

Introduction / Context:
A glass prism deviates a ray of light from its original direction. The angle of deviation depends on the angle of incidence. There is a particular condition when this deviation becomes minimum, a concept used to determine the refractive index of the prism material. This question asks you to identify the relationship between the angle of incidence and the angle of emergence at this minimum deviation condition.

Given Data / Assumptions:

• We have a triangular glass prism placed in air.
• A monochromatic ray of light enters one face of the prism and emerges from another face.
• The prism causes an overall deviation in the ray's direction, defined as the angle of deviation.
• We are specifically considering the condition of minimum deviation.

Concept / Approach:
When a prism produces minimum deviation, the path of the light ray through the prism is symmetric. In this special case, the angle of incidence (i) on the first face and the angle of emergence (e) from the second face are equal. Additionally, the angles of refraction inside the prism at the two faces are also equal. This symmetry simplifies the prism formula and is widely used in experiments to determine refractive index. The other options do not describe this symmetric condition.

Step-by-Step Solution:
Step 1: As the angle of incidence on a prism is varied, the angle of deviation first decreases, reaches a minimum value and then increases again. Step 2: At the angle of minimum deviation (denoted D_min), the light ray inside the prism travels symmetrically with respect to the prism's base. Step 3: Symmetry of the path implies that the angle of refraction at the first face equals the angle of refraction at the second face. Step 4: This symmetry further implies that the angle of incidence at the first face equals the angle of emergence at the second face. Step 5: Therefore, at minimum deviation, i = e, meaning the angle of emergence is equal to the angle of incidence.

Verification / Alternative check:
The prism formula connecting refractive index (n), prism angle (A) and minimum deviation (D_min) is n = sin((A + D_min)/2) / sin(A/2). This formula is derived under the assumption that i = e and the internal refraction angles are equal. Experimental procedures for measuring refractive index using a spectrometer specifically adjust the prism until the observed deviation is minimum and confirm that the path is symmetric by checking that rotating the telescope slightly in either direction increases the deviation. All of this confirms that equal incidence and emergence angles are characteristic of the minimum deviation case.

Why Other Options Are Wrong:
The angle of emergence is greater than the angle of incidence: This may be true for certain non-symmetric incidence values, but not at minimum deviation.

The angle of emergence is smaller than the angle of incidence: Again, this can occur for some incidence angles, but it does not correspond to the special minimum deviation condition.
The sum of the angle of incidence and the angle of emergence is exactly 90 degrees: There is no general rule that i + e = 90 degrees for a prism; this is not the defining condition for minimum deviation.

Common Pitfalls:
Students sometimes treat minimum deviation as just another incident angle rather than recognising the symmetry of the ray path. Another confusion is between internal reflection and refraction conditions. Remember that minimum deviation for a prism is directly linked to a symmetric path: equal angles of incidence and emergence and equal angles of refraction inside.

Final Answer:
At minimum deviation through a prism, the angle of emergence is equal to the angle of incidence.