Optical Path Analysis in Glass Spheres: Critical Angle for Tangential Exit

Understanding the behavior of light as it interacts with glass spheres is crucial in various fields of optics, including physics and engineering. In this article, we will explore the conditions under which a ray of light, incident on a glass sphere, can exit tangentially from the surface. We will delve into the mathematical derivations and physical principles behind this phenomenon using Snell's law and geometric considerations.

Understanding the Problem

When light travels from one medium to another, it changes direction due to the difference in the refractive indices of the two media, a phenomenon known as refraction. For a ray of light to exit a glass sphere tangentially, it must do so at the critical angle, where the refracted ray in the sphere strikes the surface at a 90-degree angle.

Relevant Physics Principles

Let's consider the following notations:

Refraction Index: n_2 3/2 for the glass sphere, and n_1 1 for air. Angle of Incidence: theta;_1 Angle of Refraction: theta;_2 Critical Angle: theta_c

Applying Snell's Law and Critical Angle

According to Snell's law, the relationship between the incident and refracted rays can be expressed as:

n_1 sintheta;_1 n_2 sintheta;_2

Condition for Tangential Exit

For the light to exit tangentially, it must meet the surface at the critical angle. The critical angle, theta_c, is defined such that:

sintheta_c n_1 / n_2

Substituting the given values:

sintheta_c 1 / (3/2) 2/3

Thus:

theta_c arcsin(2/3) approx; 41.81 degrees

Relating Angles Using Snell's Law

When light enters the sphere, it refracts according to Snell's law:

sintheta;_1 (3/2) sintheta;_2

Setting theta;_2 theta_c, we get:

sintheta;_1 (3/2) sin(arcsin(2/3))

Simplifying this expression:

sintheta;_1 (3/2) (2/3) 1

Therefore:

theta;_1 90 degrees

Conclusion

The angle of incidence (theta;_1) should be 90 degrees for the ray to exit the glass sphere tangentially. However, this solution is mathematically impossible since it violates the physical conditions of the problem. The light cannot exit along the radius; it must reflux into the sphere, making the angle of incidence and refraction re-evaluate to a practical scenario.

Given the geometric constraints, the light ray should be incident such that it makes a 90-degree angle with the radius of the glass sphere to ensure a tangential exit. In this scenario, the angle of refraction inside the sphere is arcsin(2/3), and when this ray strikes the sphere again, it refracts at 90 degrees into the air.