Two Lens System in Physics

Overview

• A problem involving a two-lens system can seem intimidating but can be simplified using the thin lens formula.
• The main idea: First, the first lens creates an image of the object, which then acts as an object for the second lens. The second lens creates the final image.
• Key question: What image does our eye see through these two lenses?

Step-by-Step Approach

Calculation with the First Lens

• Lens Type: Convex
  • Focal length, ( f_1 ) = +12 cm (positive because it's convex)
• Object Distance:
  • Measure from the center of the lens to the object.
  • Given: 24 cm + 12 cm = 36 cm
• Image Distance Calculation:
  • Thin lens formula: ( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} )
    • ( \frac{1}{12} = \frac{1}{36} + \frac{1}{d_i} )
  • Solve for ( d_i ): ( d_i = 18 ) cm (positive, on the same side as the eye)
  • The image formed by the first lens is 18 cm from the lens center.

Calculation with the Second Lens

• Image from First Lens as Object for Second Lens:
  • Distance from the lens to the image: 33 cm (total) - 18 cm = 15 cm
• Lens Type: Diverging
  • Focal length, ( f_2 ) = -10 cm (negative because it's diverging)
• Image Distance Calculation:
  • Thin lens formula: ( \frac{1}{-10} = \frac{1}{15} + \frac{1}{d_i} )
  • Solve for ( d_i ): ( d_i = -6 ) cm (negative, on the same side as the eye)

Final Image Position

• The final image is 6 cm to the left of the second lens center.

Magnification Calculations

First Lens Magnification

• Formula: ( M_1 = \frac{-d_i}{d_o} )
  • ( M_1 = \frac{-18}{36} = -\frac{1}{2} )
  • Image is 1/2 the size of the object and inverted.

Second Lens Magnification

• Formula: ( M_2 = \frac{-d_i}{d_o} )
  • ( M_2 = \frac{-(-6)}{15} = \frac{2}{5} )
  • Image is 2/5 the size of the first image and maintains orientation (still inverted).

Total Magnification

• Combined magnification: ( M_{total} = M_1 \times M_2 )
  • ( M_{total} = -\frac{1}{2} \times \frac{2}{5} = -\frac{1}{5} )
  • Final image is 1/5 the size of the original object and inverted.

Key Takeaways

• Treat each lens separately using the thin lens formula.
• Use magnification formulas to determine image size and orientation.
• Overall magnification is the product of individual magnifications.
• The thin lens formula provides image position; magnification gives size and orientation.
• An understanding of this setup is crucial for applications like microscopes.