🎢Principles of Physics II Unit 9 – Geometric Optics

Key Concepts and Terminology

• Geometric optics deals with the propagation of light using rays and assumes light travels in straight lines
• Reflection occurs when light bounces off a surface and follows the law of reflection
• Refraction happens when light bends as it passes through different media due to a change in the speed of light
• Focal point is the point where parallel rays converge after passing through a lens or reflecting off a curved mirror
• Focal length measures the distance from the center of a lens or mirror to its focal point
• Magnification describes how much larger or smaller an image appears compared to the original object
• Real image forms when light rays converge and can be projected onto a screen
• Virtual image forms when light rays appear to diverge from a point and cannot be projected onto a screen

Fundamental Laws of Geometric Optics

• Fermat's principle states that light follows the path of least time between two points
  • This explains why light rays follow straight lines in homogeneous media and bend at interfaces between different media
• Law of reflection states that the angle of incidence equals the angle of reflection when light reflects off a surface
  • The incident ray, reflected ray, and normal to the surface all lie in the same plane
• Snell's law describes how light bends when it passes from one medium to another with different refractive indices
  • The ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of the velocities in the two media, or equivalently, to the reciprocal of the ratio of the indices of refraction ($n_1 \sin \theta_1 = n_2 \sin \theta_2$)
• Total internal reflection occurs when light traveling from a medium with a higher refractive index to one with a lower refractive index is completely reflected back into the first medium
  • This happens when the angle of incidence is greater than the critical angle ($\theta_c = \sin^{-1}(n_2/n_1)$)

Reflection and Mirrors

• 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 forms
  • When the object is between the focal point and the mirror, a virtual, upright, and magnified image forms
• Convex mirrors are curved outward and always form virtual, upright, and smaller images
• Spherical aberration occurs when parallel rays away from the center of a spherical mirror do not converge at the same focal point, resulting in a blurred image
• Parabolic mirrors eliminate spherical aberration by ensuring all parallel rays converge at a single focal point

Refraction and Lenses

• Convex (converging) lenses are thicker at the center and can form real or virtual images
  • When the object is beyond the focal point, a real, inverted, and smaller or larger image forms
  • When the object is between the focal point and the lens, a virtual, upright, and magnified image forms
• Concave (diverging) lenses are thinner at the center and always form virtual, upright, and smaller images
• Lensmaker's equation relates the focal length of a lens to its refractive index and the radii of curvature of its surfaces ($1/f = (n-1)(1/R_1 - 1/R_2)$)
• Thin lens equation describes the relationship between the object distance, image distance, and focal length ($1/f = 1/d_o + 1/d_i$)
• Chromatic aberration occurs when different wavelengths of light are refracted differently by a lens, causing color fringing around the edges of an image
  • Achromatic lenses minimize this effect by combining lenses with different dispersion properties

Optical Instruments and Applications

• Cameras use a convex lens to form a real, inverted image on a light-sensitive surface (film or digital sensor)
  • The aperture and shutter speed control the amount of light entering the camera and the exposure time
• 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 large primary mirror to collect light and a smaller secondary mirror to redirect the light 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
• Fiber optics rely on total internal reflection to transmit light signals over long distances with minimal loss
  • Graded-index fibers have a refractive index that gradually decreases from the center to the edges, reducing modal dispersion

Problem-Solving Techniques

• Draw ray diagrams to visualize the path of light through optical systems
  • For mirrors, draw rays parallel to the principal axis, through the center of curvature, and through the focal point
  • For lenses, draw rays parallel to the principal axis, through the center of the lens, and through the focal point
• Use the mirror equation ($1/f = 1/d_o + 1/d_i$) to solve problems involving the object distance, image distance, and focal length of mirrors
• Use the thin lens equation ($1/f = 1/d_o + 1/d_i$) to solve problems involving the object distance, image distance, and focal length of lenses
• Apply the magnification equation ($M = -d_i/d_o = h_i/h_o$) to determine the size and orientation of images formed by mirrors and lenses
• Break complex optical systems into simpler components and analyze them step by step
  • For example, in a compound microscope, first determine the image formed by the objective lens, then use that image as the object for the eyepiece lens

Real-World Examples and Demonstrations

• Mirages are caused by refraction due to temperature gradients in the atmosphere, creating the illusion of water on a hot road or an oasis in the desert
• Rainbows form when sunlight is refracted and internally reflected within water droplets, separating the colors and creating a circular arc
• Prescription glasses use concave or convex lenses to correct vision problems such as myopia (nearsightedness) and hyperopia (farsightedness)
• Prisms can be used to disperse white light into its constituent colors, as demonstrated by Newton's famous experiment
• Optical illusions, such as the "broken pencil" illusion, rely on refraction to create a distorted image when an object is partially submerged in water
• Solar cookers use parabolic mirrors to concentrate sunlight onto a small area, generating high temperatures for cooking food
• Lighthouses use Fresnel lenses, which are made of concentric rings of prisms, to collimate light from a source and project it over long distances

Common Misconceptions and FAQs

• Misconception: Mirrors reverse left and right
  • Reality: Mirrors reverse front and back, creating the illusion of left-right reversal
• Misconception: Wearing glasses weakens your eyesight over time
  • Reality: Glasses correct vision problems but do not weaken the eyes; vision may continue to change due to other factors
• Misconception: Lasers emit a single, pure color of light
  • Reality: While many lasers emit monochromatic light, some lasers can produce multiple wavelengths or even tunable output
• FAQ: Why do we see a rainbow?
  • Rainbows appear when sunlight is refracted and reflected inside water droplets, separating the colors and creating a spectrum
• FAQ: How do 3D movies work?
  • 3D movies use polarized glasses or active shutter glasses to present slightly different images to each eye, creating the illusion of depth perception
• FAQ: What is the difference between a convex and concave lens?
  • A convex lens is thicker at the center and focuses light, while a concave lens is thinner at the center and diverges light
• FAQ: Why do stars appear to twinkle?
  • Atmospheric turbulence causes refraction of starlight, making stars appear to twinkle, while planets are less affected due to their larger apparent size