Formula Sheet: Reflection and Refraction

Reflection and Refraction Formula Sheet

Fundamental Concepts

• Speed of Light in Vacuum: \(c = 3.00 \times 10^8\) m/s
• Speed of Light in Medium: \[v = \frac{c}{n}\]
  • \(v\) = speed of light in the medium (m/s)
  • \(c\) = speed of light in vacuum (m/s)
  • \(n\) = index of refraction of the medium (dimensionless)
• Index of Refraction: \[n = \frac{c}{v}\]
  • \(n\) ≥ 1 for all materials (\(n = 1\) for vacuum)
  • Common values: air ≈ 1.00, water ≈ 1.33, glass ≈ 1.5, diamond ≈ 2.42

Law of Reflection

• Angle of Incidence equals Angle of Reflection: \[\theta_i = \theta_r\]
  • \(\theta_i\) = angle of incidence (measured from the normal)
  • \(\theta_r\) = angle of reflection (measured from the normal)
  • Both angles measured from the normal (perpendicular) to the surface
  • Incident ray, reflected ray, and normal all lie in the same plane

Snell's Law (Law of Refraction)

• General Form: \[n_1 \sin\theta_1 = n_2 \sin\theta_2\]
  • \(n_1\) = index of refraction of first medium
  • \(\theta_1\) = angle of incidence in first medium (from normal)
  • \(n_2\) = index of refraction of second medium
  • \(\theta_2\) = angle of refraction in second medium (from normal)
  • Valid at the interface between two transparent media
• Alternative Forms:
  • \(\displaystyle \sin\theta_2 = \frac{n_1}{n_2}\sin\theta_1\)
  • \(\displaystyle \frac{\sin\theta_1}{\sin\theta_2} = \frac{n_2}{n_1} = \frac{v_1}{v_2}\)
• Key Principles:
  • Light bends toward the normal when entering a denser medium (\(n_2 > n_1\))
  • Light bends away from the normal when entering a less dense medium (\(n_2 <>
  • No bending occurs when light enters perpendicular to the surface (\(\theta_1 = 0°\))

Total Internal Reflection

• Critical Angle: \[\sin\theta_c = \frac{n_2}{n_1}\]
  • \(\theta_c\) = critical angle (degrees or radians)
  • \(n_1\) = index of refraction of denser medium (where light originates)
  • \(n_2\) = index of refraction of less dense medium
  • Condition: Only occurs when \(n_1 > n_2\) (light traveling from denser to less dense medium)
  • When \(\theta_1 > \theta_c\), total internal reflection occurs (no refraction)
• Alternative Form: \[\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)\]
• Applications:
  • Fiber optics
  • Prisms in optical instruments
  • Mirages
  • Diamonds' sparkle

Plane Mirrors

• Image Distance equals Object Distance: \[d_i = -d_o\]
  • \(d_i\) = image distance (negative for virtual images)
  • \(d_o\) = object distance (always positive)
• Magnification: \[m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} = 1\]
  • \(m\) = magnification (dimensionless)
  • \(h_i\) = image height
  • \(h_o\) = object height
  • For plane mirrors, \(m = 1\) (same size, upright)
• Image Characteristics:
  • Virtual (cannot be projected on a screen)
  • Upright (same orientation as object)
  • Same size as object
  • Laterally inverted (left-right reversed)

Spherical Mirrors

Mirror Equation

• General Mirror Equation: \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
  • \(f\) = focal length
  • \(d_o\) = object distance
  • \(d_i\) = image distance
  • All distances measured from the mirror surface
• Focal Length and Radius of Curvature: \[f = \frac{R}{2}\]
  • \(R\) = radius of curvature of the mirror
  • \(f\) = focal length

Magnification for Spherical Mirrors

• Magnification Equation: \[m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}\]
  • \(m > 0\): upright image
  • \(m < 0\):="" inverted="">
  • \(|m| > 1\): enlarged image
  • \(|m| < 1\):="" reduced="">

Sign Conventions for Mirrors

• Focal Length (\(f\)):
  • Positive for concave mirrors (converging)
  • Negative for convex mirrors (diverging)
• Object Distance (\(d_o\)):
  • Always positive for real objects (in front of mirror)
• Image Distance (\(d_i\)):
  • Positive for real images (in front of mirror, same side as object)
  • Negative for virtual images (behind mirror)
• Height and Magnification:
  • Positive for upright images
  • Negative for inverted images

Spherical Lenses

Thin Lens Equation

• Lensmaker's Equation (Thin Lens): \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
  • \(f\) = focal length of lens
  • \(d_o\) = object distance from lens
  • \(d_i\) = image distance from lens
  • Same form as mirror equation but with different sign conventions

Magnification for Lenses

• Magnification Equation: \[m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}\]
  • Same formula as for mirrors
  • \(m > 0\): upright image
  • \(m < 0\):="" inverted="">

Power of a Lens

• Power Definition: \[P = \frac{1}{f}\]
  • \(P\) = power of lens (diopters, D)
  • \(f\) = focal length in meters
  • 1 diopter = 1 m^-1
  • Positive power: converging lens
  • Negative power: diverging lens
• Combined Power of Thin Lenses in Contact: \[P_{total} = P_1 + P_2 + P_3 + ...\]
  • Alternative: \(\displaystyle \frac{1}{f_{total}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + ...\)

Sign Conventions for Lenses

• Focal Length (\(f\)):
  • Positive for converging lenses (convex, biconvex)
  • Negative for diverging lenses (concave, biconcave)
• Object Distance (\(d_o\)):
  • Positive when object is on the same side as incoming light (real object)
  • Negative for virtual objects
• Image Distance (\(d_i\)):
  • Positive for real images (opposite side of lens from object)
  • Negative for virtual images (same side of lens as object)

Dispersion

• Relationship between Wavelength and Refraction:
  • Index of refraction varies with wavelength
  • \(n\) increases as wavelength decreases
  • Violet light (shorter λ) bends more than red light (longer λ)
• Applications:
  • Prisms separate white light into spectrum
  • Rainbows
  • Chromatic aberration in lenses

Optical Path Length

• Optical Path Length: \[OPL = n \times d\]
  • \(OPL\) = optical path length
  • \(n\) = index of refraction
  • \(d\) = physical distance traveled in the medium
  • Represents the equivalent distance light would travel in vacuum
• Time for Light to Travel: \[t = \frac{n \times d}{c}\]
  • \(t\) = time
  • \(d\) = distance in medium
  • \(c\) = speed of light in vacuum

Apparent Depth

• Apparent Depth Formula: \[d_{apparent} = \frac{d_{actual}}{n}\]
  • \(d_{apparent}\) = apparent depth (how deep an object appears)
  • \(d_{actual}\) = actual depth
  • \(n\) = index of refraction of the medium
  • Objects underwater appear closer to the surface than they actually are
• General Relationship: \[\frac{n_1}{d_o} + \frac{n_2}{d_i} = 0\]
  • For flat refracting surface separating two media
  • \(n_1\) = index of object's medium
  • \(n_2\) = index of observer's medium

Ray Diagrams - Key Rays

For Mirrors (Concave and Convex)

• Ray 1: Parallel to principal axis → reflects through focal point (concave) or appears to come from focal point (convex)
• Ray 2: Through focal point → reflects parallel to principal axis
• Ray 3: Through center of curvature → reflects back on itself
• Ray 4: Hits vertex → reflects at equal angle to principal axis

For Lenses (Converging and Diverging)

• Ray 1: Parallel to principal axis → passes through (or appears to come from) focal point on opposite side
• Ray 2: Through center of lens → continues straight (no deviation)
• Ray 3: Through focal point on object side → emerges parallel to principal axis

Special Cases and Important Relationships

• Object at Focal Point of Converging Lens/Mirror:
  • Image forms at infinity
  • \(\frac{1}{d_i} = 0\), so \(d_i = \infty\)
• Object at Center of Curvature (Concave Mirror):
  • Image forms at center of curvature
  • \(d_o = 2f\), then \(d_i = 2f\)
  • Magnification \(m = -1\) (inverted, same size)
• Object Very Far Away (\(d_o \to \infty\)):
  • Image forms at focal point
  • \(d_i = f\)
• Convex Mirrors and Diverging Lenses:
  • Always produce virtual, upright, reduced images
  • Never produce real images

Fermat's Principle

• Principle of Least Time:
  • Light travels along the path that takes the least time
  • Explains both reflection and refraction
  • Used to derive Snell's Law