Concept Summary

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Topic | Discussion | Learning Aids
Refraction | 26.1 THE INDEX OF REFRACTION The change in speed as a ray of light goes from one material to another causes the ray to deviate from its incident direction. This change in direction is called refraction. The index of refraction n of a material is the ratio of the speed c of light in a vacuum to the speed v of light in the material: |
Index of refraction |

| (26.1)

The values for n are greater than unity, because the speed of light in a material medium is less than it is in a vacuum.
| 26.2 SNELL’S LAW AND THE REFRACTION OF LIGHT The refraction that occurs at the interface between two materials obeys Snell’s law of refraction. This law states that (1) the refracted ray, the incident ray, and the normal to the interface all lie in the same plane, and (2) the angle of refraction is related to the angle of incidence according to |
Snell’s law of refraction |

| (26.2)

where n1 and n2 are the indices of refraction of the incident and refracting media, respectively. The angles are measured relative to the normal.
| Because of refraction, a submerged object has an apparent depth that is different from its actual depth. If the observer is directly above (or below) the object, the apparent depth (or height) d′ is related to the actual depth (or height) d according to |
Apparent depth |

| (26.3)

where n1 and n2 are the refractive indices of the materials in which the object and the observer, respectively, are located.
| 26.3 TOTAL INTERNAL REFLECTION When light passes from a material with a larger refractive index n1 into a material with a smaller refractive index n2, the refracted ray is bent away from the normal. If the incident ray is at the critical angle , the angle of refraction is 90°. The critical angle is determined from Snell’s law and is given by |
Critical angle |

| (26.4)

Total internal reflection | When the angle of incidence exceeds the critical angle, all the incident light is reflected back into the material from which it came, a phenomenon known as total internal reflection. |
Brewster angle | 26.4 POLARIZATION AND THE REFLECTION AND REFRACTION OF LIGHT When light is incident on a nonmetallic surface at the Brewster angle , the reflected light is completely polarized parallel to the surface. The Brewster angle is given by |
Brewster’s law |

| (26.5)

where n1 and n2 are the refractive indices of the incident and refracting media, respectively. When light is incident at the Brewster angle, the reflected and refracted rays are perpendicular to each other.
Dispersion | 26.5 THE DISPERSION OF LIGHT: PRISMS AND RAINBOWS A glass prism can spread a beam of sunlight into a spectrum of colors because the index of refraction of the glass depends on the wavelength of the light. Thus, a prism bends the refracted rays corresponding to different colors by different amounts. The spreading of light into its color components is known as dispersion. The dispersion of light by water droplets in the air leads to the formation of rainbows. |
Index matching | A prism will not bend a light ray at all, neither up nor down, if the surrounding fluid has the same refractive index as the glass, a condition known as index matching. |
Use Self-Assessment Test 26.1 to evaluate your understanding of Sections 26.1, 26.2, 26.3, 26.4 and 26.5 .
Focal point and focal length of a converging lens Focal point and focal length of a diverging lens Ray tracing | 26.6 LENSES 26.7 THE FORMATION OF IMAGES BY LENSES Converging lenses and diverging lenses depend on the phenomenon of refraction in forming an image. With a converging lens, paraxial rays that are parallel to the principal axis are focused to a point on the axis by the lens. This point is called the focal point of the lens, and its distance from the lens is the focal length f. Paraxial light rays that are parallel to the principal axis of a diverging lens appear to originate from its focal point after passing through the lens. The distance of this point from the lens is the focal length f. The image produced by a converging or a diverging lens can be located via a technique known as ray tracing, which utilizes the three rays outlined in the Reasoning Strategy given in Section 26.7. |
Image formed by a converging lens | The nature of the image formed by a converging lens depends on where the object is situated relative to the lens. When the object is located at a distance from the lens that is greater than twice the focal length, the image is real, inverted, and smaller than the object. When the object is located at a distance from the lens that is between the focal length and twice the focal length, the image is real, inverted, and larger than the object. When the object is located within the focal length, the image is virtual, upright, and larger than the object. |
Image formed by a diverging lens | Regardless of the position of a real object, a diverging lens always produces an image that is virtual, upright, and smaller than the object. |
| 26.8 THE THIN-LENS EQUATION AND THE MAGNIFICATION EQUATION The thin-lens equation can be used with either converging or diverging lenses that are thin, and it relates the object distance do, the image distance di, and the focal length f of the lens: |
Thin-lens equation |

| (26.6)

Magnification | The magnification m of a lens is the ratio of the image height hi to the object height ho and is also related to do and di by the magnification equation: |
Magnification equation |

| (26.7)

The algebraic sign conventions for the variables appearing in the thin-lens and magnification equations are summarized in the Reasoning Strategy given in Section 26.8.
| 26.9 LENSES IN COMBINATION When two or more lenses are used in combination, the image produced by one lens serves as the object for the next lens. |
Accommodation Near point Far point | 26.10 THE HUMAN EYE In the human eye, a real, inverted image is formed on a light-sensitive surface, called the retina. Accommodation is the process by which the focal length of the eye is automatically adjusted, so that objects at different distances produce sharp images on the retina. The near point of the eye is the point nearest the eye at which an object can be placed and still have a sharp image produced on the retina. The far point of the eye is the location of the farthest object on which the fully relaxed eye can focus. For a young and normal eye, the near point is located 25 cm from the eye, and the far point is located at infinity. |
Nearsightedness Farsightedness | A nearsighted (myopic) eye is one that can focus on nearby objects, but not on distant objects. Nearsightedness can be corrected with eyeglasses or contacts made from diverging lenses. A farsighted (hyperopic) eye can see distant objects clearly, but not objects close up. Farsightedness can be corrected with converging lenses. |
| The refractive power of a lens is measured in diopters and is given by |
Refractive power |

| (26.8)

where f is the focal length of the lens and must be expressed in meters. A converging lens has a positive refractive power, and a diverging lens has a negative refractive power.
| 26.11 ANGULAR MAGNIFICATION AND THE MAGNIFYING GLASS The angular size of an object is the angle that it subtends at the eye of the viewer. For small angles, the angular size in radians is |
Angular size |

| where ho is the height of the object and do is the object distance. The angular magnification M of an optical instrument is the angular size of the final image produced by the instrument divided by the reference angular size of the object, which is that seen without the instrument: |
Angular magnification |

| (26.9)

| A magnifying glass is usually a single converging lens that forms an enlarged, upright, and virtual image of an object placed at or inside the focal point of the lens. For a magnifying glass held close to the eye, the angular magnification M is approximately |
Angular magnification of a magnifying glass |

| (26.10)

where f is the focal length of the lens, di is the image distance, and N is the distance of the viewer’s near point from the eye.
| 26.12 THE COMPOUND MICROSCOPE A compound microscope usually consists of two lenses, an objective and an eyepiece. The final image is enlarged, inverted, and virtual. The angular magnification M of such a microscope is approximately |
Angular magnification of a compound microscope |

| (26.11)

where fo and fe are, respectively, the focal lengths of the objective and eyepiece, L is the distance between the two lenses, and N is the distance of the viewer’s near point from his or her eye.
| 26.13 THE TELESCOPE An astronomical telescope magnifies distant objects with the aid of an objective and an eyepiece, and it produces a final image that is inverted and virtual. The angular magnification M of a telescope is approximately |
Angular magnification of an astronomical telescope |

| (26.12)

where fo and fe are, respectively, the focal lengths of the objective and the eyepiece.
Spherical aberration | 26.14 LENS ABERRATIONS Lens aberrations limit the formation of perfectly focused or sharp images by optical instruments. Spherical aberration occurs because rays that pass through the outer edge of a lens with spherical surfaces are not focused at the same point as those that pass through near the center of the lens. |
Chromatic aberration | Chromatic aberration arises because a lens focuses different colors at slightly different points. |
Use Self-Assessment Test 26.2 to evaluate your understanding of Sections 26.6, 26.7, 26.8, 26.9, 26.10, 26.11, 26.12, 26.13 and 26.14 .

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