Diffraction of Light Due to a Thin Slit
Objective
To observe the phenomenon of diffraction of light due to a thin slit and determine the wavelength of the given monochromatic light source.
Apparatus Required
- Optical bench with mounting uprights
- Sodium lamp or laser source (monochromatic light source)
- Single slit of known width
- Screen (white paper or graph paper)
- Meter scale
- Optical bench with adjustable stands
- Converging lens
- Adjustable slit mechanism
- Micrometer
Theory
Diffraction is the bending of light waves around obstacles and the spreading out of light waves when they pass through narrow openings. When light passes through a single narrow slit, it forms a pattern of bright and dark bands on a screen placed at a distance.
The diffraction pattern consists of a bright central maximum flanked by alternating dark and bright fringes of decreasing intensity. The position of the dark fringes is given by:
$a \sin \theta = n\lambda$
where:
$a$ = width of the slit
$\theta$ = angle of diffraction
$n$ = order of the minima (1, 2, 3, ...)
$\lambda$ = wavelength of light
For small angles, $\sin \theta \approx \tan \theta = \frac{y}{D}$, where $y$ is the distance from the central maximum to the $n^{th}$ dark fringe, and $D$ is the distance from the slit to the screen.
Therefore, $a \cdot \frac{y}{D} = n\lambda$
Rearranging, $\lambda = \frac{a \cdot y}{n \cdot D}$
We can determine the wavelength of the light by measuring the position of the dark fringes, the slit width, and the distance to the screen.
The intensity distribution in a single-slit diffraction pattern is given by:
$$I = I_0 \left( \frac{\sin \beta}{\beta} \right)^2$$
where $\beta = \frac{\pi a \sin \theta}{\lambda}$
The minima occur when $\sin \beta = 0$, which happens when $\beta = n\pi$ (where $n = 1, 2, 3, ...$)
Substituting, we get:
$$\frac{\pi a \sin \theta}{\lambda} = n\pi$$
Which simplifies to:
$$a \sin \theta = n\lambda$$
For small angles, $\sin \theta \approx \theta \approx \frac{y}{D}$, leading to the formula:
$$\lambda = \frac{a \cdot y}{n \cdot D}$$
Experimental Setup
Fig.1: Experimental setup for diffraction through a thin slit
The experimental setup consists of a monochromatic light source (sodium lamp or laser), a single thin slit, and a screen placed at a distance. The slit width can be adjusted using a micrometer screw. The optical bench allows precise measurement of the distance between the components.
Procedure
Place the monochromatic light source at one end of the optical bench.
Mount the single slit on an adjustable stand and place it on the optical bench at a suitable distance from the light source.
Position the screen at the other end of the optical bench, ensuring it is perpendicular to the optical axis.
Adjust the positions of the components to obtain a clear diffraction pattern on the screen.
Measure and record the width of the slit ($a$) using a micrometer.
Measure and record the distance ($D$) from the slit to the screen.
On the screen, mark the position of the central maximum and the positions of the dark fringes on both sides.
Measure the distances ($y_n$) from the central maximum to each dark fringe.
Repeat the measurements for different positions of the screen (different values of $D$).
Observations
Slit width (a) = ________ mm
| Distance from slit to screen (D) in cm | Distance from central maximum to dark fringe ($y_n$) in mm | |||||
|---|---|---|---|---|---|---|
| Left 3rd ($y_{-3}$) | Left 2nd ($y_{-2}$) | Left 1st ($y_{-1}$) | Right 1st ($y_1$) | Right 2nd ($y_2$) | Right 3rd ($y_3$) | |
Calculations
Using the formula $\lambda = \frac{a \cdot y_n}{n \cdot D}$, calculate the wavelength of light for each observation.
| Order ($n$) | $y_n$ (mm) | $D$ (cm) | $a$ (mm) | $\lambda = \frac{a \cdot y_n}{n \cdot D}$ (nm) |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 |
Mean wavelength of light ($\lambda$) = ________ nm
Example calculation for the first order minimum ($n=1$):
Given:
- Slit width ($a$) = 0.05 mm
- Distance from slit to screen ($D$) = 100 cm
- Distance from central maximum to first minimum ($y_1$) = 11.7 mm
Substituting into the formula:
$$\lambda = \frac{a \cdot y_1}{1 \cdot D} = \frac{0.05 \text{ mm} \times 11.7 \text{ mm}}{1 \times 100 \text{ cm}}$$
Converting units:
$$\lambda = \frac{0.05 \times 11.7 \times 10^{-3} \times 10^{-3}}{1 \times 10^{-2}} \text{ m} = 5.85 \times 10^{-7} \text{ m} = 585 \text{ nm}$$
This is approximately the wavelength of yellow sodium light (589 nm).
Results
The wavelength of the monochromatic light source used in this experiment is ________ nm.
The standard value of the wavelength for sodium light is 589 nm (or specify for the light source used).
Percentage error = $\frac{|\text{Experimental value} - \text{Standard value}|}{\text{Standard value}} \times 100\%$ = ________%
Discussion
In this experiment, we observed the diffraction pattern formed when monochromatic light passes through a single thin slit. The pattern consists of a bright central maximum flanked by alternating dark and bright fringes. The positions of these fringes depend on:
- The wavelength of light ($\lambda$)
- The width of the slit ($a$)
- The distance from the slit to the screen ($D$)
We measured the positions of the dark fringes and used the formula $\lambda = \frac{a \cdot y_n}{n \cdot D}$ to calculate the wavelength of the light source.
The diffraction pattern observed is a result of the wave nature of light. When light waves pass through a narrow slit, they spread out and interfere with each other, creating a pattern of constructive and destructive interference on the screen.
The intensity distribution in the pattern is given by:
$$I = I_0 \left( \frac{\sin \beta}{\beta} \right)^2 \text{ where } \beta = \frac{\pi a \sin \theta}{\lambda}$$
The central maximum is the brightest and widest, with intensity $I_0$. The secondary maxima have decreasing intensities, with the first secondary maxima having only about 4.7% of the intensity of the central maximum.
The angular width of the central maximum (between the first minima on either side) is:
$$\theta_{central} = \frac{2\lambda}{a}$$
This shows that the diffraction pattern spreads out more for:
- Smaller slit widths
- Longer wavelengths
Our experimental results demonstrate this relationship and allow us to determine the wavelength of the light source based on the diffraction pattern observed.
Precautions
- Ensure that the optical bench is horizontal and stable throughout the experiment.
- The slit, light source, and screen should be properly aligned along the optical axis.
- The room should be darkened to observe the diffraction pattern clearly.
- Measure the slit width accurately using a micrometer.
- The screen should be perpendicular to the optical axis.
- Take multiple readings and average them to minimize random errors.
- Avoid parallax error while taking measurements.
- If using a laser source, never look directly into the beam as it can damage your eyes.
Conclusion
In this experiment, we successfully observed the diffraction pattern due to a single thin slit. By measuring the positions of the dark fringes and using the diffraction formula, we determined the wavelength of the monochromatic light source to be ________ nm. This value has a percentage error of ________% compared to the standard value.
The experiment demonstrates the wave nature of light and confirms the theoretical predictions of the diffraction phenomenon.
Additional Activities
- Investigate how the diffraction pattern changes when the slit width is varied.
- Study the effect of using different colored filters or light sources on the diffraction pattern.
- Compare the diffraction patterns from single, double, and multiple slits.
- Plot a graph of $y_n$ vs $n$ and determine the wavelength from the slope.