**Determination of the Wavelength of a Laser Source Using Single-Slit Diffraction**
### **Theory**
The wavelength of a laser beam can be determined using single-slit diffraction. When a monochromatic laser light passes through a narrow single slit, it undergoes diffraction, producing an interference pattern of bright and dark fringes on a screen. The central maximum is the brightest, and alternating dark and bright fringes appear on either side.
The diffraction condition for the minima is given by:
\[ a \sin \theta = m \lambda \]
Where:
- \( a \) = Width of the slit
- \( \theta \) = Angle of diffraction
- \( m \) = Order of the minimum (m = \( \pm1, \pm2, \pm3,...\))
- \( \lambda \) = Wavelength of the laser light
For small angles, \( \sin \theta \approx \theta \approx x/L \), where:
- \( x \) = Distance between the central maximum and the m-th order minimum
- \( L \) = Distance from the slit to the screen
Thus, the wavelength can be calculated as:
\[ \lambda = \frac{a x}{m L} \]
### **Apparatus**
1. Laser source
2. Single-slit apparatus
3. Screen
4. Meter scale
5. Optical bench
### **Procedure**
1. Set up the laser source so that it passes through the single slit and projects a diffraction pattern onto the screen.
2. Measure the width \( a \) of the slit.
3. Place the screen at a known distance \( L \) from the slit.
4. Observe and measure the distance \( x \) between the central maximum and the first-order minimum.
5. Use the formula \( \lambda = \frac{a x}{m L} \) to calculate the wavelength.
6. Repeat for different values of \( m \) and take the average value.
### **Sample Calculation**
Let’s assume the following experimental values:
- Width of the slit, \( a = 0.2 \) mm \( = 2.0 \times 10^{-4} \) m
- Distance from slit to screen, \( L = 1.0 \) m
- Distance between central maximum and first minimum, \( x = 2.5 \) mm \( = 2.5 \times 10^{-3} \) m
- First-order minimum \( (m = 1) \)
Using the formula:
\[ \lambda = \frac{a x}{m L} \]
\[ \lambda = \frac{(2.0 \times 10^{-4}) \times (2.5 \times 10^{-3})}{(1 \times 1.0)} \]
\[ \lambda = \frac{5.0 \times 10^{-7}}{1} \]
\[ \lambda = 500 \] nm
Thus, the calculated wavelength of the laser is approximately 500 nm, which is in the visible spectrum.
### **Precautions**
1. Ensure the laser is aligned properly with the slit.
2. Measure distances carefully to reduce errors.
3. Avoid external light sources that may interfere with observations.
4. Use a fine slit for better diffraction patterns.
5. Perform multiple measurements and take the average for accuracy.
By following this method, the wavelength of the laser light can be accurately determined.