To Observe Refraction and Lateral Deviation of a Beam of Light Incident Obliquely on a Glass Slab
Objective
To observe and understand the phenomenon of refraction and to determine the lateral displacement when a beam of light passes through a rectangular glass slab.
Materials Required
- Rectangular glass slab
- Drawing board
- White paper
- Drawing pins
- Pencil
- Scale
- Protractor
- Four optical pins
Theory
When a ray of light passes from one medium to another, it changes its direction at the boundary. This phenomenon is called refraction. The refraction of light is governed by the following laws:
- The incident ray, the refracted ray, and the normal to the interface at the point of incidence, all lie in the same plane.
- The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media. This is known as Snell's law, which is mathematically expressed as:
$$\frac{\sin i}{\sin r} = n_{21}$$
Where:
- $i$ is the angle of incidence
- $r$ is the angle of refraction
- $n_{21}$ is the refractive index of the second medium with respect to the first medium
When a ray of light passes through a glass slab, it undergoes refraction twice:
- First, when it enters the glass slab from air (at point of incidence)
- Second, when it emerges from the glass slab into air (at point of emergence)
Due to these two refractions, the emergent ray is parallel to the incident ray but laterally displaced from it. This displacement is called lateral displacement or lateral shift.
Lateral Displacement
The lateral displacement ($d$) of a ray of light passing through a rectangular glass slab is given by:
$$d = t \left(\frac{\sin(i-r)}{\cos r}\right)$$
Where:
- $t$ is the thickness of the glass slab
- $i$ is the angle of incidence
- $r$ is the angle of refraction
Alternatively, it can also be expressed as:
$$d = t \sin i \left(\frac{1-\cos(i-r)}{\cos r}\right)$$
To derive the formula for lateral displacement, consider a ray of light passing through a glass slab:
Let's denote:
- Angle of incidence: $i$
- Angle of refraction: $r$
- Thickness of glass slab: $t$
From geometric considerations:
$$d = AB \sin(i-r)$$
Where $AB$ is the distance traveled by the ray inside the glass slab. We can calculate $AB$ as:
$$AB = \frac{t}{\cos r}$$
Substituting this value:
$$d = \frac{t}{\cos r} \sin(i-r)$$
$$d = t \left(\frac{\sin(i-r)}{\cos r}\right)$$
This is the formula for lateral displacement through a glass slab.
Procedure
- Place a white sheet of paper on the drawing board and fix it with drawing pins.
- Place the rectangular glass slab in the middle of the paper and draw its outline ABCD using a pencil.
- Remove the glass slab.
- Draw a normal NN' at point O on the side AB of the drawn outline.
- Draw a line PO at an angle (angle of incidence) with the normal.
- Place the glass slab back on its outline.
- Fix two pins P and Q on the incident ray PO.
- Look through the opposite face of the glass slab (CD) and fix two more pins R and S such that they appear to be in the same straight line as the images of pins P and Q as seen through the glass slab.
- Remove the pins and the glass slab.
- Mark the positions of the pins and join points R and S, extending the line backwards to meet CD at point O'.
- Join O and O' to obtain the path of the light ray through the glass slab.
- Extend RS backwards to meet the extended PO at point E.
- Measure OE, which gives the lateral displacement.
- Repeat the procedure for different angles of incidence and record your observations.
Diagram
Figure 1: Path of light ray through a rectangular glass slab showing refraction and lateral displacement
Observations
Record your measurements in the following table:
| S.No. | Angle of Incidence ($i$) | Angle of Refraction ($r$) | Thickness of Glass Slab ($t$) | Measured Lateral Displacement ($d_{measured}$) | Calculated Lateral Displacement ($d_{calculated}$) | Percentage Error (%) |
|---|---|---|---|---|---|---|
| 1 | ||||||
| 2 | ||||||
| 3 | ||||||
| 4 | ||||||
| 5 |
Calculations
For each observation, calculate the lateral displacement using the formula:
$$d = t \left(\frac{\sin(i-r)}{\cos r}\right)$$
Calculate the percentage error using:
$$\text{Percentage Error} = \left|\frac{d_{measured} - d_{calculated}}{d_{calculated}}\right| \times 100\%$$
Let's work through an example calculation:
Suppose we have:
- Angle of incidence ($i$) = 45°
- Angle of refraction ($r$) = 28° (calculated using Snell's law with n = 1.5)
- Thickness of glass slab ($t$) = 2.0 cm
Step 1: Calculate the lateral displacement using the formula:
$$d = t \left(\frac{\sin(i-r)}{\cos r}\right)$$
$$d = 2.0 \left(\frac{\sin(45°-28°)}{\cos 28°}\right)$$
$$d = 2.0 \left(\frac{\sin 17°}{\cos 28°}\right)$$
$$d = 2.0 \left(\frac{0.2924}{0.8829}\right)$$
$$d = 2.0 \times 0.3313$$
$$d = 0.6626 \text{ cm}$$
Step 2: If the measured lateral displacement was 0.68 cm, calculate the percentage error:
$$\text{Percentage Error} = \left|\frac{0.68 - 0.6626}{0.6626}\right| \times 100\%$$
$$\text{Percentage Error} = \left|\frac{0.0174}{0.6626}\right| \times 100\%$$
$$\text{Percentage Error} = 0.0263 \times 100\%$$
$$\text{Percentage Error} = 2.63\%$$
Results
- Plot a graph between the angle of incidence ($i$) and lateral displacement ($d$).
- Determine the refractive index of the glass slab using Snell's law: $n = \frac{\sin i}{\sin r}$
- Calculate the average value of the refractive index from all observations.
Discussion
In this section, discuss your observations and answer the following questions:
- How does the lateral displacement vary with the angle of incidence?
- Why is the emergent ray parallel to the incident ray but laterally displaced?
- What would happen to the lateral displacement if the thickness of the glass slab is increased?
- What are possible sources of error in this experiment?
- How would the lateral displacement change if we used a different material with a higher refractive index?
Conclusion
Summarize your findings and what you learned from this experiment about the refraction of light through a glass slab and the factors affecting lateral displacement.
Precautions
- The glass slab should have plane parallel faces and should be free from scratches.
- The pins should be fixed vertically.
- The eye should be positioned such that the bases of all pins appear to be in a straight line when viewed through the glass slab.
- Parallax errors should be avoided when marking the positions of pins.
- The outline of the glass slab should be drawn carefully to ensure accurate measurements.
- The normal should be drawn perpendicular to the face of the glass slab.