---
title: "A point source of light is placed at the bottom surface of a flat, transparent glass plate of thickness \\(t\\) and index of refraction \\(n\\). The top surface of the plate is in contact with air, which has an index of refraction of approximately \\(1.0\\). Light rays travel outward in all upward directions from the source. Light that strikes the top interface at an angle less than the critical angle escapes into the air, forming a bright circular spot of radius \\(R\\) on the top surface. Light striking the interface at an angle greater than the critical angle undergoes total internal reflection."
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url: "https://nerd-notes.com/ubq/117687/"
date_modified: "2026-08-04T07:53:14+00:00"
---

# A point source of light is placed at the bottom surface of a flat, transparent glass plate of thickness \(t\) and index of refraction \(n\). The top surface of the plate is in contact with air, which has an index of refraction of approximately \(1.0\). Light rays travel outward in all upward directions from the source. Light that strikes the top interface at an angle less than the critical angle escapes into the air, forming a bright circular spot of radius \(R\) on the top surface. Light striking the interface at an angle greater than the critical angle undergoes total internal reflection.

A point source of light is placed at the bottom surface of a flat, transparent glass plate of thickness \(t\) and index of refraction \(n\). The top surface of the plate is in contact with air, which has an index of refraction of approximately \(1.0\). Light rays travel outward in all upward directions from the source. Light that strikes the top interface at an angle less than the critical angle escapes into the air, forming a bright circular spot of radius \(R\) on the top surface. Light striking the interface at an angle greater than the critical angle undergoes total internal reflection.

![A 2D cross-section of a rectangular block representing the glass plate. A solid black dot is located on the bottom edge, centered horizontally, labeled 'Point Source'. A vertical dashed line extends upward from the point source to the top edge of the block. Two solid arrows representing light rays originate from the point source. One ray angles upward and to the left, and the other upward and to the right. Both rays terminate at the top edge of the block. A vertical dashed normal line is drawn where the right ray intersects the top edge. The angle between the right ray and this normal line is labeled \(\theta_c\). The horizontal distance along the top edge from the center dashed line to the right ray's intersection point is indicated by a bracket and labeled \(R\). The vertical thickness of the block is indicated by a vertical bracket on the left side and labeled \(t\). The area inside the block is labeled \(n\), and the area above the block is labeled 'Air'. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829994-H2K5R9.jpg)

**Part a)** **Derive** an expression for the critical angle \(\theta_c\) at the glass-air interface. Express your answer in terms of \(n\) and fundamental constants. *(2 points)*

**Part b)** **Calculate** the radius \(R\) of the bright spot on the top surface of the glass plate if the glass has an index of refraction \(n = 1.60\) and a thickness \(t = 3.00 \text{ cm}\). *(2 points)*

**Part c)** A clear liquid with an index of refraction \(n_{liq}\) (where \(1.0 < n_{liq} < n\)) is now poured evenly over the top surface of the glass plate, replacing the air at the boundary. *(3 points)*

**Part d)** **Derive** an algebraic expression for the new radius \(R_{liq}\) of the bright spot when the liquid is present. Express your answer in terms of \(t\), \(n\), \(n_{liq}\), and fundamental constants. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117687/*