Optics 2 - Refraction of Light PDF

7.4 Optics 7.4.2 Refraction Refraction at Plane Surfaces A Incident Ray N Normal Air (less dense) Angle of...

7.4 Optics 7.4.2 Refraction Refraction at Plane Surfaces A Incident Ray N Normal Air (less dense) Angle of X incidencece Y O Water (more dense) 𝜃 D Figure 1 B Where: XY = Boundary separating the two media (air and water) AO = incident ray OB = refracted ray ON = normal (the imaginary line that is drawn at right angles to the boundary.) Angle 𝑖 = angle of incidence Angle 𝑟 = angle of refraction Angle 𝜃 = angle of deviation where 𝜃 = 𝑖 − 𝑟 O = Point of incidence (It is the point where the incident ray, the refracted ray and the normal all meet.) When light travels from a less dense to a denser medium (e.g. from air to glass/water, the refracted ray bends towards the normal. L. Bonello Page 1  7.4 Optics (more dense) 𝑟 (less dense) 𝜃 Figure 2 In figure 2, the angle of deviation 𝜃 = 𝑟 − 𝑖 Note: The angle of deviation is the angle through which the light has changed its direction. When light travels from a denser to a less dense medium (e.g. from glass/water to air, the refracted ray bends away from the normal. Laws of Refraction 1. The refracted ray is in the same plane as the incident ray and the normal, at the point of incidence but on the opposite side of the normal from the incident ray. 2. For two particular media, the ratio of the sine of the angle of sin 𝑖 incidence to the sine of the angle of refraction is constant, i. e. is sin 𝑟 constant. This is known as Snell’s Law. L. Bonello Page 2 7.4 Optics Refraction of Water Waves (or Monochromatic Light) When water waves pass from deep to shallow, the wavelength changes and so does the direction of travel of the wave. This can be seen in the diagram below. 𝜆 𝜆 Figure 3 In figure 3, the top part of the incoming wave slows down before the bottom part does. This causes the wave to change direction. Note: Deep water → less dense Shallow water → more dense From deep to shallow → wave bends towards the normal The same occurs when monochromatic light (light of only one colour, one wavelength) passes from one medium to another medium (e.g., from air to water).This can be seen in figure 4. There is also a change in direction and speed. L. Bonello Page 3 7.4 Optics Figure 4 Note: Air→ less dense Water → more dense From less dense to more dense → wave bends towards the normal As the waves travel from deep water to shallow water (or from less dense to denser medium): 1. Waves slow down ∴ velocity decreases. 2. Wavelengths decrease (since 𝑣 = 𝑓𝜆 and so 𝑣 ∝ 𝜆). 3. Part of the wave travels faster for longer causing the wave to turn, i.e., change direction. 4. Frequency remains the same. Note: Frequency → Characteristic of wave source Wavelength → Characteristic of the medium L. Bonello Page 4 7.4 Optics In other words, When light travels from one medium to another, there is a change in: 1. Direction 2. Velocity 3. Wavelength BUT frequency of light remains the same. In figure 5, light moves faster in air than in glass because air is a gas and the particles are very spaced out from each other when compared to the particles of glass which is a solid and whose particles are tightly packed together. Figure 5 In figure 6, light moves slowly in glass because glass is a solid and its particles are very densely packed together when compared to the particles of air which are very far apart, noting that air is a gas. L. Bonello Page 5 7.4 Optics Figure 6 7.4.3 Refractive index The refractive index of a medium is the ratio of the velocity of light in air to the velocity of light in the medium. Some examples of refractive indices are given in the table below: Medium Refractive Index Vacuum 1.00 Air (gas) 1.00028 Water (liquid) 1.33 Glass (solid) 1.5 Table 1 L. Bonello Page 6 7.4 Optics Note: 1. Going down the table, refractive index 𝑛 increases as density of material 𝜌 increases. So, going down the table again, light travels slower due to the presence of more particles of the medium being present and the greater the bending towards the normal. 2. The refractive index of air is 1.00028. However, we normally take the refractive index of air to be equal to 1.00, i.e., the refractive index of a vacuum. Absolute refractive index of air 𝑛𝑎𝑖𝑟 = 1.00 Equations for Refractive Index 1. If the media containing the incident and refracted rays are denoted by 1 (air) and 2 (glass) (see Figure 6), then the refractive index is given by: 𝑛2 sin 𝑖 1 𝑛2 = = 𝑛1 sin 𝑟 (Snell’s Law) where: 1𝑛2 = refractive index of second medium (glass) with respect to the first medium (air). (𝑁𝑜 𝑢𝑛𝑖𝑡𝑠) 𝑛2 = refractive index of second medium (e.g., glass in fig. 6) (𝑁𝑜 𝑢𝑛𝑖𝑡𝑠) 𝑛1 = refractive index of air (𝑛1 = 1) (𝑁𝑜 𝑢𝑛𝑖𝑡𝑠) 𝑖 = angle of incidence (°) 𝑟 = angle of refraction (°) L. Bonello Page 7 7.4 Optics E. g. A ray of light is incident on an air – glass boundary, making an angle of incidence of 50°. What is the angle of refraction if the refractive index of glass is 1.5? Air Glass Figure 7 Using Snell’s Law: sin 𝑖 1𝑛2 = sin 𝑟 sin 50° 1.5 = sin 𝑟 sin 50° sin 𝑟 = 1.5 sin 50° 𝑟 = sin−1 ( ) 1.5 0.7660 𝑟 = sin−1 ( ) 1.5 𝑟 = sin−1 0.5107 𝑟 = 30.7 ° Note: Use this form of Snell’s Law when the first medium is 𝐧𝟐 𝐬𝐢𝐧 𝐢 air. If first medium is not air, then use =. 𝐧𝟏 𝐬𝐢𝐧 𝐫 L. Bonello Page 8 7.4 Optics 2. Also, 𝑣1 1𝑛2 = 𝑣2 Where: 1𝑛2 = refractive index of second medium with respect to the first medium (No units since 1𝑛2 is a ratio.) 𝑣1 = velocity of light in medium 1 (𝑚 𝑠 −1 ) 𝑣2 = velocity of light in medium 2 (𝑚 𝑠 −1 ) Note: The word absolute is used when the first medium is a vacuum, then, 1𝑛2 = absolute refractive index of second medium with respect to the first medium which is a vacuum. E. g. If the speed of light in air is 3 𝑥 108 𝑚 𝑠 −1 and the refractive index of diamond is 2.42, then work out the speed of light passing through the diamond. 𝑣1 1𝑛2 = 𝑣2 3 𝑥 108 2.42 = 𝑣2 3 𝑥 108 𝑣2 = 2.42 𝑣2 = 1.24 𝑥 108 𝑚𝑠 −1 L. Bonello Page 9 7.4 Optics 𝑣1 3. Also, since 1𝑛2 = …Equation 1 𝑣2 And 𝑣 = 𝑓𝜆 …Equation 2 Then, substituting equation 2 in equation 1: 𝑓𝜆1 1𝑛2 = 𝑓𝜆2 𝜆1 1𝑛2 = 𝜆2 where: 1𝑛2 = refractive index of second medium with respect to the first medium (No units since 1𝑛2 is a ratio.) 𝜆1 = wavelength of light in medium 1 (𝑚) 𝜆2 = wavelength of light in medium 2 (𝑚) E.g., If the refractive index of glass for blue light is 1.55 and the wavelength of blue light in air is 4.5 𝑥 10−7 𝑚, what will be the wavelength of blue light in glass? 𝜆1 1𝑛2 = 𝜆2 4.5 𝑥 10−7 1.55 = 𝜆2 4.5 𝑥 10−7 𝜆2 = 1.55 𝜆2 = 2.90 𝑥 10−7 𝑚 L. Bonello Page 10 7.4 Optics 4. Also, when looking into water or glass, an object appears to be closer to the observer. E.g., an object at the bottom of a swimming pool appears to be closer than it actually is as seen in figure 8. Figure 8 𝑟𝑒𝑎𝑙 𝑑𝑒𝑝𝑡ℎ 𝐴𝑂 𝑛= = 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑑𝑒𝑝𝑡ℎ 𝐴𝐼 E.g., Robert finds his goggles at the bottom of a swimming pool which is 2.5 m deep. If the refractive index of water is 1.33, find the apparent depth of the goggles. 𝑟𝑒𝑎𝑙 𝑑𝑒𝑝𝑡ℎ 𝑛= 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑑𝑒𝑝𝑡ℎ 2.5 1.33 = 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑑𝑒𝑝𝑡ℎ 2.5 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑑𝑒𝑝𝑡ℎ = 1.33 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑑𝑒𝑝𝑡ℎ = 1.88 𝑚 L. Bonello Page 11 7.4 Optics 5.. We can also write Snell’s Law more symmetrically: From the equation on page 7: 𝑛2 sin 𝑖 = 𝑛1 sin 𝑟 𝑛2 sin 𝜃1 = (𝑖 can be written as 𝜃1 and 𝑟 can be written as 𝜃2 ) 𝑛1 sin 𝜃2 Cross-multiply: 𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2 𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2 where: 𝑛1 = refractive index of medium 1 (𝑁𝑜 𝑢𝑛𝑖𝑡𝑠) 𝑛2 = refractive index of medium 2 (𝑁𝑜 𝑢𝑛𝑖𝑡𝑠) 𝜃1 = angle of incidence in medium 1 (°) 𝜃2 = angle of refraction in medium 2 (°) E. g. Light hits a water - glass boundary at an angle of 60°. What is the angle of refraction in glass if the refractive indices for water and glass are 1.33 and 1.54, respectively. Water Glass Figure 9 L. Bonello Page 12 7.4 Optics Using Snell’s Law: 𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2 1.33 sin 60° = 1.54 sin 𝜃2 1.33 sin 60° sin 𝜃2 = 1.54 1.33 𝑥 0.8660 sin 𝜃2 = 1.54 sin 𝜃2 = 0.7479 ∴ 𝜃2 = sin−1 0.7479 ∴ 𝜃2 = 48.4 ∴ 𝜃2 = 48° Note: This more symmetrical form of the equation for Snell’s Law can be used for any pair of media. The Principle of Reversibility of Light This states that the paths of light are reversible. Considering the worked example above: 𝜃1 in water = 60° and 𝜃2 in glass = 48° when light is travelling from water to glass. If, on the other hand, light travels from glass to water, then: 𝜃1 in glass = 48 ° and 𝜃2 in water = 60°. Hence, the ray simply retraces its original path in the reverse direction. L. Bonello Page 13