Algebraic Relationship Between Object Distance, Image Distance, and Focal Length

Introduction

• The goal is to establish an algebraic relationship between:
  • Distance of the object from the convex lens.
  • Distance of the image from the convex lens.
  • The focal length of the lens.

Setup

• Green object represents the object to be imaged.
• Pink points represent the focal points.
• Rays drawn:
  • Parallel ray from the tip of the arrow, refracted through the focal point on the opposite side.
  • Ray through the focal point on the same side, refracted parallel.
• Image characteristics:
  • Inverted
  • Real
  • Larger than the object

Labeled Distances

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

Methodology

Similar Triangles Approach

1. Identify Similar Triangles:

   • Use geometry to find similar triangles by angle comparisons.
   • Identify corresponding sides.

2. Establish Ratios:

   • For similar triangles:
     • Ratio of corresponding sides is equal.
     • Example: ( \frac{d₀}{dᵢ} = \frac{A}{B} )

Relationship Between Variables

• Aim is to relate these triangles and focal length to derive a formula.

Detailed Steps

• Triangles and Corresponding Sides:
  1. Identify similar triangles using:
     • Opposite angles (vertical angles).
     • Alternate interior angles (due to parallel lines).
  2. Formulate ratios of corresponding sides.
  3. Utilize these ratios to find relationships between distances:
     • ( \frac{d₀}{dᵢ} = \frac{A}{B} )
     • ( \frac{A}{B} = \frac{f}{dᵢ - f} )

Algebraic Derivation

1. Combine Ratios:
   • If ( \frac{d₀}{dᵢ} = \frac{f}{dᵢ - f} ), equate them.
2. Cross-multiply to simplify:
   • Simplification process involves basic algebra.
3. Resulting Equation:
   • ( \frac{1}{f} = \frac{1}{d₀} + \frac{1}{dᵢ} )

Conclusion

• Derived a clean and elegant formula for convex lenses:
  • Relates focal length, object distance, and image distance.
  • Formula: ( \frac{1}{f} = \frac{1}{d₀} + \frac{1}{dᵢ} )
• Demonstrates a successful application of geometry and algebra to optics.