25.3 The Law of Refraction

Light bends when it moves between different materials. This bending, called refraction, follows Snell's law. Understanding refraction helps explain everyday phenomena like why objects appear bent in water or how rainbows form.

The speed of light changes in different materials, which affects how much it bends. This relationship is described by the index of refraction. Knowing how light behaves in various materials is key to understanding optical devices and natural light phenomena.

The Law of Refraction

Angle of refraction calculation

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• Snell's law relates angle of incidence ($\theta_1$) and angle of refraction ($\theta_2$) when light passes through boundary between two media with different indices of refraction ($n_1$ and $n_2$)
  • Snell's law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$
  • Example: Light passing from air ($n_1 = 1$) to water ($n_2 = 1.33$)
• Calculate angle of refraction:
  1. Determine indices of refraction for two media ($n_1$ and $n_2$)
  2. Measure angle of incidence ($\theta_1$) relative to normal line
  3. Rearrange Snell's law to solve for $\theta_2$: $\theta_2 = \arcsin(\frac{n_1}{n_2} \sin \theta_1)$
• Light bends towards normal line when traveling from lower to higher index of refraction medium (air to water)
• Light bends away from normal line when traveling from higher to lower index of refraction medium (water to air)
• Snell's law is named after Willebrord Snellius, who formulated it in 1621

Light speed in materials

• Speed of light in vacuum ($c$) approximately $3 \times 10^8$ m/s
• In any other medium, speed of light ($v$) slower than in vacuum
• Index of refraction ($n$) of medium is ratio of speed of light in vacuum to speed of light in that medium: $n = \frac{c}{v}$
  • Higher index of refraction indicates slower speed of light in medium (diamond, $n = 2.42$)
  • Lower index of refraction indicates faster speed of light in medium (air, $n = 1$)
• Speed of light in medium related to its optical density
  • Optically denser media have higher indices of refraction and slower light speeds (glass)
  • Optically less dense media have lower indices of refraction and faster light speeds (air)

Applications of Snell's law

• Identify two media involved and their respective indices of refraction ($n_1$ and $n_2$)
• Determine angle of incidence ($\theta_1$) and whether angle of refraction ($\theta_2$) or index of refraction of second medium ($n_2$) is unknown
• If angle of refraction unknown:
  • Use Snell's law to calculate $\theta_2$: $\theta_2 = \arcsin(\frac{n_1}{n_2} \sin \theta_1)$
  • Example: Light passing from air ($n_1 = 1$) to water ($n_2 = 1.33$) at 30° angle of incidence
• If index of refraction of second medium unknown:
  • Rearrange Snell's law to solve for $n_2$: $n_2 = \frac{n_1 \sin \theta_1}{\sin \theta_2}$
  • Example: Light passing from air to unknown medium at 45° angle of incidence, refracted at 30°
• Special cases:
  • Total internal reflection occurs when light travels from higher to lower index of refraction medium at critical angle ($\theta_c$)
    • Critical angle: $\theta_c = \arcsin(\frac{n_2}{n_1})$
    • Example: Light traveling from water to air at angles greater than 48.8°
  • No refraction at normal incidence (0°), light continues in straight line

Additional Optical Phenomena

• Dispersion: Separation of white light into its component colors due to wavelength-dependent refraction
• Frequency of light remains constant during refraction, while wavelength changes in different media
• Polarization: Process by which light waves are restricted to vibrate in a single plane