Thin Lens Formula and Sign Conventions

Thin Lens Equation

• Formula: ( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} )
• f: focal length
  • Distance from lens center to focal point.
  • Positive for converging (convex) lenses.
  • Negative for diverging (concave) lenses.
• d_o: object distance
  • Distance from lens center to object.
  • Always positive for a single lens.
• d_i: image distance
  • Distance from lens center to image.
  • Positive if image is on the opposite side of the lens from the object.
  • Negative if on the same side as the object.

Focal Length Determination

• Converging/Convex Lens
  • Focal length is positive.
  • Example: 8 cm measured gives +8 cm.
• Diverging/Concave Lens
  • Focal length is negative.
  • Example: 8 cm measured gives -8 cm.

Object and Image Distance

• Object Distance (d_o)
  • Always positive if only using one lens.
  • Example: 30 cm from lens center is +30 cm.
• Image Distance (d_i)
  • Positive if on opposite side as the object, same side as the eye looking through the lens.
  • Negative if on the same side as the object, opposite side of the eye.

Sign Conventions and Usage

• Eye Placement
  • Eyes should look through the lens at the object.
  • Image distance positive if closer to eye than object.
  • Negative image distance means image is on the same side as the object.
• Magnification (M) Formula
  • Formula: ( M = -\frac{d_i}{d_o} )
  • Positive magnification: right-side up image.
  • Negative magnification: inverted image.

Example Problem

• Given a concave (diverging) lens with 8 cm focal length:
  • Focal length ( f = -8 ) cm.
  • Object distance ( d_o = 24 ) cm.
  • Solve for ( d_i ) using the formula, resulting in ( d_i = -6 ) cm.
  • Negative ( d_i ) indicates image on the same side as object.
  • Magnification calculation gives positive value, confirming right-side up image.
  • Image is one-fourth the size of the object (e.g., if object height is 8 cm, image height is 2 cm).

Additional Notes

• The thin lens formula provides only horizontal distances.
• To calculate image height, use the magnification formula.