Lecture Notes: Young's Double Slit Experiment

Introduction

• Problem: Measuring path length difference between two very close slits (micrometers/nanometers apart).
• Challenge: Physically measuring the path length difference is difficult due to closeness of holes.

Solution Overview

• Determine path length difference based on angle.
• Use a reference line through the center to measure angles.

Methodology

• Reference Line: Draw a line through the center to simplify angle measurement.
• Angle Measurement: Measure angle from centerline to a point on the wall.
• Path Length Difference: Use angles to determine path length difference.

Geometric Setup

• Screen Position: The screen is far away from the slits.
• Use lines from the center of each slit to the point on the wall.
• Right Angle: Third line is drawn perpendicular to the other two to form a right angle.

Trigonometric Analysis

• Right Triangle: Formed with:
  • Distance between holes (d).
  • Path length difference ( ( \Delta x ) ).
• Angles: Two angles are equal due to the setup.

Calculating Path Length Difference

• Trigonometric Relationship:
  • ( \sin(\theta) = \frac{\Delta x}{d} )
  • ( \Delta x ) (path length difference) = ( d \times \sin(\theta) )

Double Slit Formula

• Formula: ( M \times \lambda = d \times \sin(\theta) )
• Variables:
  • ( M ): Order of the constructive point (0, 1, 2, ...).
  • ( \lambda ): Wavelength of the light.
  • ( \theta ): Angle from centerline to constructive point.
  • ( d ): Distance between the slits.

Constructive and Destructive Interference

• Constructive Points: When ( \Delta x = M \lambda )
• Destructive Points: Can occur when ( \Delta x = (M + 0.5) \lambda )

Practical Application

• Measure theta and known wavelength to find distance ( d ).
• Use diffraction patterns to figure out slit spacing.
• Useful for determining spacing in crystal lattices or molecular structures using Young's Double Slit Equation.

Conclusion

• Even if ( d ) is very small and hard to measure, using angles and known wavelengths allows for accurate determination of slit spacing.