Lenses and mirrors are key players in geometric optics. They bend and reflect light, creating images we use every day. From eyeglasses to telescopes, these optical elements shape how we see the world.

Understanding how lenses and mirrors work is crucial for grasping geometric optics. We'll explore their types, properties, and how they form images. This knowledge forms the foundation for more complex optical systems and applications.

Converging vs Diverging Lenses and Mirrors

Optical Properties and Characteristics

• Converging lenses and mirrors focus parallel light rays to a single point while diverging lenses and mirrors spread parallel light rays outward
• Converging lenses appear thicker at the center than at the edges while diverging lenses appear thinner at the center than at the edges
• Converging mirrors have a concave reflective surface while diverging mirrors have a convex reflective surface
• Focal point of converging lenses and mirrors locates in front of the optical element (real) while for diverging lenses and mirrors locates behind the optical element (virtual)
• Converging lenses and mirrors form both real and virtual images whereas diverging lenses and mirrors only form virtual images
  • Real image forms when light rays actually intersect (projectable on a screen)
  • Virtual image forms when light rays appear to intersect but do not actually meet

Sign Conventions and Image Formation

• Focal length sign convention uses positive for converging lenses and mirrors and negative for diverging lenses and mirrors
• Converging lenses and mirrors can produce both enlarged and reduced images depending on object distance
  • Objects beyond 2f produce reduced real images
  • Objects between f and 2f produce enlarged real images
  • Objects within f produce enlarged virtual images
• Diverging lenses and mirrors always produce reduced virtual images
  • Virtual images appear closer to the optical element than the object

Focal Length and Radius of Curvature

Relationships and Calculations

• Focal length (f) of a spherical mirror relates to its radius of curvature (R) by the equation $f = R/2$
• Concave mirrors have positive focal length and radius of curvature while convex mirrors have negative values
• Focal length determined experimentally by finding the convergence point of parallel light rays after reflection
• Radius of curvature measured directly or calculated from focal length using $R = 2f$
• Smaller radius of curvature results in more strongly curved mirror surface and shorter focal length
  • Example: A mirror with R = 20 cm has f = 10 cm, while a mirror with R = 50 cm has f = 25 cm

Paraxial Approximation and Applications

• Large radii of curvature allow for paraxial approximation simplifying calculations and ray diagrams
• Paraxial approximation assumes small angles and thin lenses, valid when object and image are close to the optical axis
• Applications of spherical mirrors include telescopes (concave) and security mirrors (convex)
• Focal length affects magnification and image characteristics in optical systems
  • Shorter focal lengths produce greater magnification (microscopes)
  • Longer focal lengths produce less distortion (telescopes)

Image Location with Ray Diagrams

Principal Rays and Image Formation

• Ray diagrams for spherical mirrors use three principal rays
  • Ray parallel to principal axis reflects through focal point
  • Ray through center of curvature reflects back on itself
  • Ray through focal point reflects parallel to principal axis
• Concave mirrors form real images when object locates beyond focal point and virtual images when object locates between focal point and mirror
• Convex mirrors only form virtual images appearing behind mirror and smaller than object
• Image position determined by intersection point of two or more reflected rays
• Ray diagrams predict image characteristics
  • Real or virtual nature
  • Upright or inverted orientation
  • Relative size compared to object

Ray Diagram Accuracy and Applications

• Accuracy of ray diagrams increases with number of rays traced but three principal rays usually suffice for most applications
• Ray diagrams provide visual representation of image formation process
• Applications include designing optical systems (cameras, telescopes) and understanding image characteristics in different mirror types
• Practice drawing ray diagrams enhances understanding of image formation principles
  • Example: Draw ray diagram for object at 2f distance from concave mirror
  • Example: Compare ray diagrams for convex and concave mirrors with same focal length

Mirror Equation and Magnification Problems

Mirror Equation and Sign Conventions

• Mirror equation relates object distance (do), image distance (di), and focal length (f) $\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$
• Sign convention for distances in mirror equation
  • Positive for real objects and images (in front of mirror)
  • Negative for virtual objects and images (behind mirror)
• Convex mirrors always have negative focal length in calculations
• Magnification equation for mirrors $m = -\frac{d_i}{d_o}$
  • Negative magnification indicates inverted image
  • Positive magnification indicates upright image

Problem-Solving Strategies

• Combine mirror equation and magnification equation to calculate image characteristics (size, position, orientation) given object characteristics and mirror properties
• These equations apply to both concave and convex mirrors with proper attention to sign conventions
• Problem-solving steps
  1. Identify given information and unknown variables
  2. Choose appropriate equation (mirror equation or magnification equation)
  3. Substitute known values and solve for unknown
  4. Check answer for reasonableness and proper signs
• Examples of calculations
  • Calculate image distance for object 30 cm from concave mirror with 20 cm focal length
  • Determine magnification of convex mirror with 15 cm focal length for object 40 cm away