Overview

This experiment determines the refractive index of water by shining a narrow laser beam vertically onto a shallow dish of water and measuring the diameter of the circular light ring formed due to total internal reflection at the water-air interface.

What This Experiment Demonstrates

• Refraction of light at the water surface
• Total internal reflection in a shallow water layer
• Relationship between ring diameter, water depth, and refractive index

Apparatus Required

• Shallow dish or tray
• Water
• Narrow beam laser pointer
• White screen or paper to project the ring
• Ruler or measuring scale
• Pencil and paper for recording measurements

Theory

When a laser beam enters water vertically, light rays hitting the water-air interface at an angle greater than the critical angle undergo total internal reflection. The result is a **circular light ring observed at the bottom or projected on a screen.

If the depth of water is \( h \) and the radius of the ring formed is \( r \), then the refractive index \( \mu \) is given by:

\[ \mu = \frac{r}{\sqrt{4h^2 + r^2}}\]

This formula comes from trigonometry, using the relationship between the critical angle \( \theta_c \) and the dimensions of the light path:

\[ \sin \theta_c = \frac{h}{\sqrt{h^2 + (r/2)^2}} \quad , \quad \mu = \frac{1}{\sin \theta_c} \]

Method

1. Place the shallow dish on a white sheet or screen.
2. Fill the dish with water to a known depth \( h \).
3. Direct the laser pointer vertically down into the center of the dish.
4. Observe the circular light ring formed on the bottom of the dish or on the screen.
5. Measure the radius \( r \) of the ring using a ruler.
6. Repeat measurements for different water depths if desired.
7. Use the formula \( \mu = \frac{r}{\sqrt{4h^2 + r^2}}\) to calculate the refractive index of water.

Observations

Record the following:

• Water depth \( h \) (cm)
• Ring radius \( r \) (cm)
• Calculated refractive index \( \mu \)

Calculations

Using the measured values:

\[ \mu = \frac{r}{\sqrt{4h^2 + r^2}}\]

Take multiple readings for accuracy and calculate the mean refractive index:

\[ \mu_\text{mean} = \frac{\mu_1 + \mu_2 + \mu_3 + \dots}{n} \]

Result

The refractive index of water is obtained using the measured ring radius and water depth, confirming the refraction properties of water and total internal reflection.

Sources of Error & Precautions

• Ensure the laser is perfectly vertical; tilt causes distorted ring.
• Maintain a still water surface to avoid ripples affecting ring diameter.
• Measure the ring radius accurately using a ruler.
• Repeat multiple trials for accurate averaging.
• Avoid parallax while measuring the ring diameter.

Video Demonstration

Video courtesy of Turbulent Tech World (YouTube). Used for educational purposes under YouTube’s embedding policy.