Lecture on Convex Lenses and Algebraic Relationships

Introduction

• Overview of convex lenses and their role in forming images.
• Goal: Develop an algebraic relationship between:
  • Distance of the object from the lens
  • Distance of the image from the lens
  • Focal length of the lens

Diagram Explanation

• The object is represented by a green shape.
• Focal points (pink) are a focal length away from the lens.
• Ray tracing:
  • Parallel ray from the object refracts through the focal point.
  • Ray through the focal point refracts and becomes parallel.
• Intersection of refracted rays provides the image location.
  • Inverted, real image, larger than the object.

Developing the Relationship

• Labeling Distances:

  • d₀ (Object Distance): Distance from object to lens.
  • dᵢ (Image Distance): Distance from image to lens.
  • f (Focal Length): Distance from lens to focal point.

• Using Geometry:

  • Identify similar triangles to derive relationships.
  • Triangle Analysis:
    • Draw and flip triangles for comparison.
    • Use angles and parallel lines to prove similarity.
    • Ratio of sides for similar triangles.

• Formulas Derived from Triangles:

  • Ratio of object distance to image distance equals ratio of arbitrary lengths in similar triangles.
  • Relating arbitrary lengths to focal length through similar triangle analysis.

Algebraic Derivation

• Start from similarity ratios to derive:
  • d₀/dᵢ = a/b = f/(dᵢ - f)
• Simplify through algebra:
  • Cross multiply and simplify.
  • Factor and separate terms.
  • Final form: 1/f = 1/d₀ + 1/dᵢ

Conclusion

• Achieved algebraic relationship for convex lenses:
  • Relates focal length to object/image distances.
  • Simplified, neat formula.
• Significance: Provides a clean method to determine image formation parameters for convex lenses.