**Introduction:** The purpose of this experiment is to determine the index of refraction for various materials using different techniques. There are 3 parts to this experiment.

- Apply
*Snell’s law*to determine the index of refraction for a clear slab of plastic (acrylic). - Use the critical angle to determine the index of refraction for a slab of plastic.
- Use Pfund’s method to determine the index of refraction for ‘soda-lime’ glass slab.

## Part 1: Snell’s Law

**Procedure:** A single-slit, white light source will be directed through a material whose index of refraction is to be determined. By laying the slab of material on a piece of paper and visually observing the path of the light source as it enters and exits the slab, we can measure the angles of incidence (entering) and refraction (exiting).

Using those 2 values (theta1 and theta2) and the known index of refraction for air (1.00) we can compute the index of refraction using Snell’s law: sin(theta1)*n1=sin(theta2)*n2

Where:

n1=1.00

theta1=angle of incidence

theta2=angle of refraction

n2=index of refraction for material

Solving for n2 and simplifying we derive the following equation: **n2=sin(theta1)/sin(theta2)**

Ten separate measurements were taken at varying angles between 0 and 90 degrees. The path of the light source was also noted.

### Setup:

### Data and Calculations:

n2=sin(theta1)/sin(theta2)

**Conclusion:** By applying Snell’s Law, we were able to determine the approximate index of refraction to be 1.458 with a standard error of 0.027. This is very close to the known value of 1.490. There are several potential sources of error in this experiment which include the following:

- An uncertainty in the measurement of the angle by at least 0.5 degrees.
- A “shaky” set-up may cause an offset when drawing the path of the light before the angle is even measured.
- The material may have imperfections affecting the path of light.

## Part 2: Using Critical Angle

**Procedure:** Using a semicircular slab of material (acrylic), we are to determine the critical angle through experiment and use that value to calculate the index of refraction. The critical angle is the angle at which light will no longer ‘pass through’ a material and instead be totally reflected.

To find the critical angle, a light ray is aimed through one side of the flat end of a plastic semicircle and adjusted to allow total internal reflection to occur until the light finally exits the plastic slab from the other side of the flat part, nearly antiparallel to the original projection.

From this, we can measure a quantity, R which is the distance from where the first reflection occurs to the ‘center’ of the slab, a ‘radius’ away. A second reflection occurs before exiting the slab, using the value of R that we measure, we can determine a value L, which is the distance from the edge of the slab to where the light exits. Applying simple laws of geometry allow us to calculate the angle that the rays reflect, which is the critical angle.

### Setup:

### Data and Calculations:

(light region)

(dark region)

= asin((R-L)/R)

(n1)/(sin(criticalangle))

R

L

diameter

radius

critical angle

Refractive Index

5.1

1.25

7.6

3.8

49.01680406

1.324675325

#### Conclusion:

By experimentally measuring the critical angle of a ray of light through a medium, we were able to calculate a relatively close value of the material’s refractive index.

Some of the uncertainties involved were due to the requirement of very precise measurements (to the millimeter) while using crude devices, ie., a cheap plastic ruler, etc. Another area of error may be in attempting to precisely locate the border between the light and dark region since it was not a very sharp contrast.

Overall, the end measurement appears to be very close to the actual index of refraction for that material.

## Part 3: Pfund’s Method

**Intro/Procedure:** Our adaptation of Pfund’s Method includes a thick glass plate with a paint coated on one side. A laser beam is aimed at the plate with an incident angle of 0 degrees. The laser then reflects off the back surface of the slab (where there is a coating of paint) and diffusely reflects back toward the incident surface. As long as the diffused rays do not reach a critical angle they will continue to pass through the surface. To determine the critical angle, the setup is manipulated to create a sharp ring on the painted glass, this indicates the critical angle has been reached.

By measuring the diameter of this ring along with the thickness of the slab, one can calculate the index of refraction.

### Setup:

### Data and Calculations:

asin((D/4)/sqrt(T^2+(D/4)^2))

(n1)/(sin(criticalangle))

diameter (D)

thickness (T)

Critical Angle

Refractive Index

3.5

0.97

42.05241392

1.492960352

**Conclusion**: Using Pfund’s method, we were able to very accurately determine the index of refraction for the glass slab. The main source of error was simply the precision of measurements which were relatively negligible based on the results.