🧲AP Physics 2 (2025) Unit 13 – Geometric Optics

Key Concepts and Terminology

• Geometric optics deals with the study of light as rays that travel in straight lines and interact with various surfaces and media
• Light rays are idealized representations of light that travel in straight lines and change direction when they encounter a boundary between two different media
• Reflection occurs when light rays bounce off a surface (mirror) and change direction according to the law of reflection
• Refraction happens when light rays bend as they pass from one medium to another with a different optical density (water, glass)
• Focal point is the point where light rays converge after passing through a lens or reflecting off a curved mirror
  • Real focal point is where light rays actually converge and can be projected onto a screen
  • Virtual focal point is where light rays appear to converge but do not actually meet
• Image formation depends on the interaction of light rays with mirrors and lenses, resulting in real or virtual images
• Magnification refers to the ratio of the size of an image to the size of the object, which can be greater than, equal to, or less than 1

Reflection and Mirrors

• Law of reflection states that the angle of incidence equals the angle of reflection when light reflects off a surface
• Plane mirrors produce virtual, upright, and laterally inverted images that appear to be the same distance behind the mirror as the object is in front
• Concave mirrors are curved inward and can form real or virtual images depending on the object's distance from the mirror
  • When the object is beyond the focal point, a real, inverted, and smaller image is formed
  • When the object is between the focal point and the mirror, a virtual, upright, and magnified image is formed
• Convex mirrors are curved outward and always form virtual, upright, and smaller images
• Spherical aberration occurs when light rays from different parts of a spherical mirror do not converge at the same point, causing image distortion
• Parabolic mirrors minimize spherical aberration by ensuring that all light rays parallel to the principal axis converge at the focal point

Refraction and Lenses

• Refraction occurs when light passes from one medium to another with a different refractive index, causing the light ray to bend
• Snell's law relates the angles of incidence and refraction to the refractive indices of the media: $n_1 \sin \theta_1 = n_2 \sin \theta_2$
• Total internal reflection happens when light traveling from a higher to a lower refractive index medium reaches a critical angle and is completely reflected back into the original medium
• Lenses are optical devices that use refraction to focus or diverge light rays
  • Converging lenses (convex) focus light rays and can form real or virtual images
  • Diverging lenses (concave) spread light rays and always form virtual, upright, and smaller images
• Thin lens equation relates the object distance ($d_o$), image distance ($d_i$), and focal length ($f$) of a lens: $\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$
• Lens maker's equation relates the focal length of a lens to its refractive index ($n$) and the radii of curvature of its surfaces ($R_1$ and $R_2$): $\frac{1}{f} = (n - 1)(\frac{1}{R_1} - \frac{1}{R_2})$

Optical Instruments and Applications

• Cameras use a converging lens to form a real, inverted image on a light-sensitive surface (film or digital sensor)
  • Aperture controls the amount of light entering the camera and affects depth of field
  • Shutter speed determines the duration of light exposure and affects motion blur
• Telescopes use a combination of lenses or mirrors to magnify distant objects
  • Refracting telescopes use a large objective lens to collect light and a smaller eyepiece lens to magnify the image
  • Reflecting telescopes use a primary mirror to collect light and a secondary mirror to redirect it to the eyepiece
• Microscopes use a combination of lenses to magnify small objects
  • Compound microscopes have an objective lens that forms a real, magnified image, which is then further magnified by the eyepiece lens
  • Scanning electron microscopes (SEM) use a focused beam of electrons to produce high-resolution images of sample surfaces
• Fiber optics use total internal reflection to transmit light signals over long distances with minimal loss
  • Used in telecommunications, internet infrastructure, and medical imaging (endoscopes)
• Prisms disperse white light into its constituent colors due to the wavelength dependence of refractive index (dispersion)

Mathematical Models and Equations

• Mirror equation relates the object distance ($d_o$), image distance ($d_i$), and focal length ($f$) of a mirror: $\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$
• Magnification equation expresses the ratio of image height ($h_i$) to object height ($h_o$) in terms of image and object distances: $m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$
• Thin lens equation (see Refraction and Lenses section)
• Lens maker's equation (see Refraction and Lenses section)
• Power of a lens is the reciprocal of its focal length in meters: $P = \frac{1}{f}$, measured in diopters (D)
• Snell's law (see Refraction and Lenses section)
• Critical angle equation relates the critical angle ($\theta_c$) to the refractive indices of the media: $\sin \theta_c = \frac{n_2}{n_1}$, where $n_1 > n_2$

Experimental Techniques and Lab Work

• Determine the focal length of a converging lens by focusing a distant object (e.g., the sun) and measuring the image distance
• Verify the thin lens equation by measuring object and image distances for various object positions and calculating the focal length
• Investigate the relationship between the angle of incidence and the angle of reflection using a plane mirror and a protractor
• Demonstrate total internal reflection using a laser pointer and a semicircular glass block
• Measure the refractive index of a material (e.g., glass or water) using Snell's law and a refraction experiment setup
  • Use a laser pointer, a semicircular block of the material, and a protractor to measure the angles of incidence and refraction
• Construct a simple telescope or microscope using converging lenses and observe the magnification and image properties
• Observe chromatic dispersion using a prism and a white light source, and measure the angles of deviation for different colors

Common Misconceptions and Pitfalls

• Confusing the terms "reflection" and "refraction" or their associated phenomena
• Believing that an object's distance from a plane mirror affects the size of the image
• Thinking that a concave mirror always produces a real, inverted image, regardless of the object's position
• Assuming that a converging lens always forms a real image, even when the object is within the focal length
• Misinterpreting the sign conventions for object and image distances in the mirror and thin lens equations
• Forgetting to consider the refractive indices of both media when applying Snell's law
• Neglecting to account for the wavelength dependence of refractive index (dispersion) in optical systems
• Misunderstanding the role of aperture and shutter speed in camera settings and their effects on image quality

Real-World Applications and Examples

• Mirrors are used in various applications, such as rearview mirrors in vehicles, dental mirrors, and solar concentrators
• Lenses are essential components in cameras, telescopes, microscopes, and corrective eyewear (glasses and contact lenses)
• Fiber optic cables enable high-speed internet communication and are used in medical imaging devices (endoscopes)
• Prisms are used in spectrometers to analyze the composition of light and in binoculars to invert images
• Fresnel lenses are used in lighthouse lamps to collimate light and increase visibility for ships
• Retroreflectors, which reflect light back to its source, are used in road signs, bicycle reflectors, and lunar laser ranging experiments
• Gradient-index (GRIN) lenses, which have a varying refractive index, are used in fiber optic coupling and miniature imaging systems
• Adaptive optics systems, which use deformable mirrors or liquid crystal spatial light modulators, are used in astronomy to correct for atmospheric distortion