Chapter 26 Geometrical Optics

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Chapter 26 Units of Chapter 26 • The Reflection of Light Geometrical Optics • Forming Images with a Plane Mirror • Spherical Mirrors • Rayyg Tracing and the Mirror E quation • The Refraction of Light • Ray Tracing for Lenses • The Thin-Lens Equation • Dispersion and the Rainbow

26-1 The Reflection of Light 26-1 The Reflection of Light If a stone is dropped into a pond, circular Spherical waves and plain waves waves emanate from the point where it landed. Rays, perpendicular to the wave fronts, give the direction in which the waves propagate.

26-1 The Reflection of Light 26-1 The Reflection of Light

The law of reflection states that the angle of Reflection from a smooth surface is called incidence equals the angle of reflection: specular reflection; if the surface is rough, it is diffuse reflection.

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26-2 Forming Images with a Plane Mirror 26-2 Forming Images with a Plane Mirror

Light reflected from the flower and vase hits the Properties of Mirror Images Produced by Plane mirror. Obeying the law of reflection, it enters Mirrors: the eye. The eye interprets the ray as having had a straight-line path, and sees the image • A mirror image is upright, but appears reversed behind the mirror. right to left. • A mirror image appears to be the same distance behind the mirror that the object is in front of the mirror. • A mirror image is the same size as the object.

26-2 Forming Images with a Plane Mirror 26-3 Spherical Mirrors

A corner reflector reflects light A spherical mirror has the shape of a parallel to the incident ray, no section of a sphere. If the outside is matter the incident angle. mirrored, it is convex; if the inside is mirrored, it is concave.

What’s the minimum height of mirror in order to see the full image of yours?

26-3 Spherical Mirrors 26-3 Spherical Mirrors Parallel rays hitting a spherical mirror come together at the focal point (or appear to have come from the focal point, if the mirror is convex).

For a convex mirror, the focal length is negative, as the rays do not go through the focal point. The opposite is true for a concave mirror.

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26-3 Spherical Mirrors 26-4 Ray Tracing and the Mirror Equation We have made the assumption here that the rays do not hit the mirror very far from the principal axis. If they do, the We use three principal rays in finding the image image is blurred; this is called spherical aberration, and can produced by a concave mirror. be remedied by using a parabolic mirror instead. • The parallel ray (P ray) reflects through the focal point. • The focal ray (F ray) reflects parallel to the axis. • The center-of-curvature rayy( (C ray) reflects back alon g its incoming path.

When the Hubble Space Telescope was first launched, its optics were marred by spherical aberration. This was fixed with corrective optics.

26-4 Ray Tracing and the Mirror Equation 26-4 Ray Tracing and the Mirror Equation

This image shows how these three rays are used to find The process is similar for a concave mirror, the image formed by a convex mirror. The image is although there are different results depending located where the projections of the three rays cross. The size of the image can also be determined. on where the object is placed.

26-4 Ray Tracing and the Mirror Equation Exam 2 We derive the mirror equation using the ray 100 88 66 Score Distribution 100 84 66 diagrams: 12 100 84 65 11 96 83 64 10 95 81 59 8 8 93 78 57 7 6 93 76 55 6

92 76 52 udents in Range 4 t 4 90 74 45 3 90 73 42 2 11 90 70 40 No. of S 00 0 89 70 36 0 102030405060708090 89 69 28 Score Range 88 67

Average: 74.5

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26-4 Ray Tracing and the Mirror Equation 26-4 Ray Tracing and the Mirror Equation

Using the similar triangles and the fact that f = ½ R, we get the mirror equation:

Here, do is the distance from the mirror to the object, di is the distance from the mirror to the image, and f is the focal length.

We can also find the magnification:

26-4 Ray Tracing and the Mirror Equation 26-5 The Refraction of Light Light moves at different speeds through different media. Here are the sign conventions for concave and convex When it travels from one medium into another, the mirrors: change in speed causes the ray to bend.

The angle of refraction is related to the different speeds: Example: A convex spherical mirror with a radius of curvature of 10.0 cm produces a virtual image one-third the size of the real object. Where is the object?

26-5 The Refraction of Light 26-5 The Refraction of Light

The speed of light in a medium We can now write the angle of refraction in is given by the index of terms of the index of refraction: n=1.80 refraction of that medium:

Remembering that the speed of a wave is v = ƒ λ, one wonders whether the frequency or the wavelength of light changes upon entering a medium? Here are some typical indices of refraction:

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26-5 The Refraction of Light Total Internal Reflection and Critical Angle

Refraction can make objects immersed in water appear • Total internal reflection can broken, and can create mirages. occur when light attempts to move from a medium with a high index of refraction to one with a lower index of refraction • A particular angle of incidence will result in an angle of refraction of 90°. This angle of incidence is called the critical angle • For angles of incidence greater than the critical angle, the beam is entirely reflected at the boundary

n2 sinθc = for n1 > n2 n1

Fiber Optics Refraction in a Prism

• An application of internal • The amount the ray is bent away from reflection its original direction is called the • Plastic or glass rods are used angle of deviation, δ to “pipe” light from one place • Since all the colors have different to another angles of deviation, they will spread out into a spectrum • Applications include • A prism spectrometer is commonly – MdilMedical use o ffibf fiber op tic ca bles used to study wavelengths emitted by for diagnosis and correction of a light source medical problems – Telecommunications

Example: An optical fiber consists of a transparent core (n=1.55) surrounded by cladding (n=1.35), as shown in the figure below. A ray of light in air is incident at the entrance to

the fiber with incident angle φ1. (a) What is the speed of light in the fiber core? (b) What is the

maximum value for φ1 so that the light will not refract into the cladding?

Rainbow Thin Lenses • A thin lens consists of a piece of glass or other primary rainbow transparent substance that can be characterized by the effect it has on light, mostly either converging or diverging the incoming light. • Lenses are commonly used to form images by refraction in optical instruments

• Sun light shines on distant raindrops or mist (small water spheres). • Red is observed higher (outside) than violet, with other colors in between. • These angles specify conditions when the refracted/reflected light is most intense (total internal reflection). (a) Converging lenses (b) Diverging lenses

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Focal Length of Thin Lenses 26-6 Ray Tracing for Lenses The three principal rays for lenses are similar to those for mirrors: • The focal length, ƒ, is the image distance that corresponds to an infinite object distance • The P ray—or parallel ray—approaches the lens parallel to its axis. • A thin lens has two focal points, corresponding to parallel • The F ray is drawn toward (concave) or through (convex) the focal rays from the left and from the right point. • The midpoint ray (M ray) goes through the middle of the lens. Assuming the lens is thin enough, it will not be deflected. This is the thin-litilens approximation.

26-6 Ray Tracing for Lenses 26-7 The Thin-Lens Equation

A concave lens always forms a virtual image. A convex We derive the thin-lens lens forms different image types depending on where the equation in the same way we object is located with respect to the focal point. did the mirror equation, using these diagrams:

26-7 The Thin-Lens Equation Examples Sign conventions for thin lenses: A lens produces a real image that is twice as large as the object and is located 15 cm from the lens. Find (a) the object distance and (b) the focal length of the lens.

Combinations of Thin Lenses

• The image of the first lens is treated as the object of the second lens • The image formed by the second lens is the final image of the system • The overall magnification is the product of the magnification of the separate lenses

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Summary of Chapter 26 Summary of Chapter 26 • A wave front is a surface along which the wave • Spherical mirrors have spherical reflecting phase is constant. Rays, perpendicular to the surfaces. A concave mirror is curved inward, and wave fronts, indicate the direction of a convex one outward. propagation. • Focal length of a convex mirror: • The angle of incidence equals the angle of reflection. • Focal length of a concave mirror: • The image formed by a plane mirror is upright, • An image is real if light passes through it, but appears reversed left to right; appears to be virtual if it does not. the same distance behind the mirror as the • Mirror equation: object is in front of it; and is the same size as the object.

Summary of Chapter 26 Summary of Chapter 26

• Magnification: • Snell’s law: • Light entering a medium of higher n is bent towards the normal; light entering a medium of • Refraction is the change in direction of light lower n is bent away from the normal. due to a change in speed. • When li ght ent ers a medi um of l ower n, there i s • The index of refraction gives the speed of light a critical angle beyond which the light will be in a medium: totally reflected.

Summary of Chapter 26 Summary of Chapter 26

• At Brewster’s angle, the reflected light is totally • Magnification: polarized:

• The index of refraction varies with frequency; different frequencies of light are bent different • A lens uses refraction to bend light and form amounts. This is called dispersion. images. • Thin-lens equation: